{"id":29792215,"url":"https://github.com/p1xt/markdown-notes-calculus","last_synced_at":"2026-02-09T15:03:53.915Z","repository":{"id":305551707,"uuid":"1023162631","full_name":"P1xt/markdown-notes-calculus","owner":"P1xt","description":null,"archived":false,"fork":false,"pushed_at":"2025-07-20T17:54:16.000Z","size":17,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":0,"default_branch":"main","last_synced_at":"2025-07-20T19:31:18.665Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":null,"has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":null,"status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/P1xt.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":null,"code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null}},"created_at":"2025-07-20T16:53:03.000Z","updated_at":"2025-07-20T17:54:19.000Z","dependencies_parsed_at":"2025-07-20T19:41:26.256Z","dependency_job_id":null,"html_url":"https://github.com/P1xt/markdown-notes-calculus","commit_stats":null,"previous_names":["p1xt/markdown-notes-calculus"],"tags_count":null,"template":false,"template_full_name":null,"purl":"pkg:github/P1xt/markdown-notes-calculus","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/P1xt%2Fmarkdown-notes-calculus","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/P1xt%2Fmarkdown-notes-calculus/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/P1xt%2Fmarkdown-notes-calculus/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/P1xt%2Fmarkdown-notes-calculus/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/P1xt","download_url":"https://codeload.github.com/P1xt/markdown-notes-calculus/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/P1xt%2Fmarkdown-notes-calculus/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":267447459,"owners_count":24088602,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","status":"online","status_checked_at":"2025-07-27T02:00:11.917Z","response_time":82,"last_error":null,"robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2025-07-28T01:05:36.278Z","updated_at":"2026-02-09T15:03:48.884Z","avatar_url":"https://github.com/P1xt.png","language":null,"funding_links":[],"categories":[],"sub_categories":[],"readme":"# **Chapter 1**\n\nSummary, created after skimming TOC and Guide Questions, reading the chapter, filling out my definitions list, and taking notes on the main concepts of this module:\nIn order to solve problems involving limits and continuity, I should understand what a limit is, what continuity is, and understand the 9 theorems presented in this module. Were I to make flash cards, I'd do so for all terms in the definitions list, and for each of the theorems.\n\n\u003c!-- TOC start (generated with https://github.com/derlin/bitdowntoc) --\u003e\n\n- [**Chapter 1**](#chapter-1)\n  - [**First look**](#first-look)\n    - [**Contents page (page iii)**](#contents-page-page-iii)\n    - [**Questions to guide your review (page 103)**](#questions-to-guide-your-review-page-103)\n    - [**Summary of big ideas**](#summary-of-big-ideas)\n  - [**Definitions I want to fill out by the end of the chapter**](#definitions-i-want-to-fill-out-by-the-end-of-the-chapter)\n  - [**Explanation of formal definition of a limit**](#explanation-of-formal-definition-of-a-limit)\n    - [**Plain English Explanation:**](#plain-english-explanation)\n    - [**Formal (Symbolic) Definition:**](#formal-symbolic-definition)\n    - [**What It Really Means:**](#what-it-really-means)\n    - [**Example:**](#example)\n  - [**Important theorems**](#important-theorems)\n    - [**Theorem 1**](#theorem-1)\n      - [**Rules**](#rules)\n    - [**Theorem 2**](#theorem-2)\n    - [**Theorem 3**](#theorem-3)\n      - [**How to Eliminate Zero Denominators Algebraically**](#how-to-eliminate-zero-denominators-algebraically)\n    - [**Theorem 4**](#theorem-4)\n      - [**Extensions of the Limit Concept**](#extensions-of-the-limit-concept)\n        - [**One-Sided Limits**](#one-sided-limits)\n        - [**Infinite Limits**](#infinite-limits)\n    - [**Theorem 5: Relationship Between One-Sided and Two-Sided Limits**](#theorem-5-relationship-between-one-sided-and-two-sided-limits)\n  - [**Continuity**](#continuity)\n    - [**Basic Terms**](#basic-terms)\n    - [**Continuity at a Point**](#continuity-at-a-point)\n      - [**Removable Discontinuity**](#removable-discontinuity)\n      - [**Jump Discontinuity**](#jump-discontinuity)\n      - [**Infinite Discontinuity**](#infinite-discontinuity)\n      - [**Continuous at a Left or Right Endpoint**](#continuous-at-a-left-or-right-endpoint)\n      - [**Continuity on Intervals**](#continuity-on-intervals)\n  - [**Key Theorems About Continuity**](#key-theorems-about-continuity)\n    - [**Theorem 6: Continuity of Algebraic Combinations**](#theorem-6-continuity-of-algebraic-combinations)\n    - [**Theorem 7: Continuity of Polynomials and Rational Functions**](#theorem-7-continuity-of-polynomials-and-rational-functions)\n    - [**Theorem 8: Continuity of Composite Functions**](#theorem-8-continuity-of-composite-functions)\n    - [**Theorem 9: The Intermediate Value Theorem (IVT)**](#theorem-9-the-intermediate-value-theorem-ivt)\n\n\u003c!-- TOC end --\n\n---\n\n\u003c!-- TOC --\u003ca name=\"first-look\"\u003c/a\n## **First look**\n\n\u003c!-- TOC --\u003e\u003ca name=\"contents-page-page-iii\"\u003e\u003c/a\u003e\n### **Contents page (page iii)**\nChapter 1 is about Limits and Continuity introduces:\n- rates of change\n- limits\n- rules for finding limits\n- target values\n- the formal definition of a limit\n- extensions to the limit concept\n- continuity\n- tangent lines\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"questions-to-guide-your-review-page-103\"\u003e\u003c/a\u003e\n### **Questions to guide your review (page 103)**\nReading through the questions, I see that the following concepts will be important:\n- knowing how to find the average rate of change\n- knowing how to decide what limit to calculate\n- being able to figure out whether what happens at a particular point affect the existence of and value of a limit\n- knowing what theorems are there to help easily find limits\n- knowing what one sided limits are\n- knowing how to control the input of a function to scope the limit\n- explaining limit notation in words\n- knowing what a continuous function is and what conditions have to be present for a function to be continuous\n- being able to look at a graph and tell where it is continuous\n- what does left continous and right continuous mean and how do you figure them out\n- being able to figure continuity out for a variety of functions like:\n    - polynomials\n    - trig functions\n    - rational powers\n    - combinations of functions\n    - composites of functions\n    - absolute value of functions\n- be able to explain continuity on an interval\n- be able explain what it means for a function to be continuous\n- know what properties does a function need to have to be continuous on an interval\n- what does it mean for a line to be tangent to a curve at a specific point\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"summary-of-big-ideas\"\u003e\u003c/a\u003e\n### **Summary of big ideas**\nThe main theme for this chapter is limits and continuity. By the end of it I should understand what limits are, know the main theorms for solving them, understand what continuity of a function is and why it's important to understand continuity and how it relates to limits, and understand how to figure out limits for a variety of functions.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"definitions-i-want-to-fill-out-by-the-end-of-the-chapter\"\u003e\u003c/a\u003e\n## **Definitions I want to fill out by the end of the chapter**\n\nI'll fill these out as I go through the chapter\n\n| Term                                 | Definition                                                                                                                                                                                                                                  |\n| ------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n| **rates of change**                  | The average rate of change for a function is how much the output (y) changes compared to the input (x); essentially the slope between two points.                                                                                           |\n| **limit**                            | A limit tells us what value a function is getting close to as the input (x) gets close to a certain number.                                                                                                                                 |\n| **limit notation**                   | $\\lim_{x \\to x_0} f(x) = L$ means “as $x$ gets close to $x_0$, the function $f(x)$ gets close to the value $L$.”                                                                                                                            |\n| **rules for finding limits**         | These are shortcuts and properties—like the sum, difference, product, quotient, and power rules—that help us calculate limits more easily without needing to graph.                                                                         |\n| **one-sided limits**                 | A one-sided limit looks at what value a function approaches from only one side—either from the left ($x \\to x_0^-$) or from the right ($x \\to x_0^+$).                                                                                      |\n| **target values**                    | The value that a function approaches (the “target”) as the input gets close to a specific number, regardless of what the function actually equals there.                                                                                    |\n| **the formal definition of a limit** | Uses ε (epsilon) and δ (delta) to precisely define what it means for a function to get close to a value: for every small distance (ε) you want between $f(x)$ and the limit, there is a small range (δ) around $x_0$ where this holds true. |\n| **extensions to the limit concept**  | Includes ideas like infinite limits (function grows without bound), limits at infinity (what happens as $x$ becomes very large or small), and limits involving piecewise functions or discontinuities.                                      |\n| **continuity**                       | A function is continuous at a point if the limit exists at that point, the function is defined there, and the limit equals the actual function value—meaning there are no jumps, holes, or breaks.                                          |\n| **intervals**                        | A continuous stretch of numbers on the x-axis where the function behaves a certain way; limits can be analyzed over open, closed, or infinite intervals.                                                                                    |\n| **tangent lines**                    | A tangent line touches a curve at one point and shows the direction the curve is heading at that exact point; its slope is found using a limit of the average rate of change.                                                               |\n\n---\n\u003c!-- TOC --\u003e\u003ca name=\"explanation-of-formal-definition-of-a-limit\"\u003e\u003c/a\u003e\n## **Explanation of formal definition of a limit**\n\n\u003c!-- TOC --\u003e\u003ca name=\"plain-english-explanation\"\u003e\u003c/a\u003e\n### **Plain English Explanation:**\n\nThe **formal definition of a limit** is a precise way to say:\n\n\u003e “As $x$ gets really, really close to a number, $f(x)$ gets really, really close to a specific value.”\n\nBut how close is “really close”? And how can we **prove** it?\n\nThat’s where **epsilon (ε)** and **delta (δ)** come in:\n\n* **ε (epsilon)** is how close you want $f(x)$ to get to the limit value (think vertical closeness).\n* **δ (delta)** is how close you need to make $x$ to the target number (horizontal closeness) to get that result.\n\nThe formal definition says:\n\n\u003e If you want to make the function’s output $f(x)$ as close as you like to the limit $L$, then you can always find a small distance $\\delta$ such that if $x$ is within $\\delta$ units of the target number $x_0$, then $f(x)$ will be within $\\varepsilon$ units of $L$.\n\nThink of it as a **promise**:\nIf someone picks an ε, saying \"I want $f(x)$ to be within this range of the limit,\"\nyou can find a δ such that, \"Sure—then just make sure $x$ is this close to $x_0$, and we’re good.\"\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"formal-symbolic-definition\"\u003e\u003c/a\u003e\n### **Formal (Symbolic) Definition:**\n\nWe say\n\n$$\n\\lim_{x \\to x_0} f(x) = L\n$$\n\n**if and only if**:\n\nFor every $\\varepsilon \u003e 0$, there exists a $\\delta \u003e 0$ such that\nwhenever $0 \u003c |x - x_0| \u003c \\delta$, it follows that $|f(x) - L| \u003c \\varepsilon$.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"what-it-really-means\"\u003e\u003c/a\u003e\n### **What It Really Means:**\n\n* $x$ is **not equal** to $x_0$, but very close.\n* $f(x)$ is **not required** to be equal to $L$ at $x_0$—just very close to $L$ when $x$ is close to $x_0$.\n* This definition works even when the function **isn't defined** at $x_0$, or if there’s a **hole** in the graph.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"example\"\u003e\u003c/a\u003e\n### **Example:**\n\nLet’s say\n\n$$\n\\lim_{x \\to 2} (3x + 1) = 7\n$$\n\nIf someone says:\n\n“I want $f(x)$ to stay within 0.01 of 7 (so $\\varepsilon = 0.01$),”\n\nYou can reply:\n\n“Then keep $x$ within 0.0033 of 2 (so $\\delta = 0.0033$), and it’ll work.”\n\nThis back-and-forth guarantees that no matter how tight the output range is, we can always make the input close enough to hit it.\n\n\u003c!-- TOC --\u003e\u003ca name=\"important-theorems\"\u003e\u003c/a\u003e\n## **Important theorems**\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-1\"\u003e\u003c/a\u003e\n### **Theorem 1**\n\u003c!-- TOC --\u003e\u003ca name=\"rules\"\u003e\u003c/a\u003e\n### **Rules**\n\n\n| Rule                       | Explanation                                                                                                                                               |\n| -------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------- |\n| **Sum Rule**               | The limit of a sum is the sum of the limits. In other words, you can take the limit of each part separately and then add them together.                   |\n| **Difference Rule**        | The limit of a difference is the difference of the limits. Just take the limit of each part and subtract.                                                 |\n| **Product Rule**           | The limit of a product is the product of the limits. Take the limit of each factor and then multiply the results.                                         |\n| **Constant Multiple Rule** | If you're multiplying a function by a constant, you can pull the constant out of the limit. Then just multiply the constant by the limit of the function. |\n| **Quotient Rule**          | The limit of a quotient is the quotient of the limits—as long as the limit of the denominator isn’t zero.                                                 |\n| **Power Rule**             | If a function is raised to a power, you can take the limit first, then raise the result to that power.                                                    |\n\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-2\"\u003e\u003c/a\u003e\n### **Theorem 2**\n\n**Limits of polynomials can be found by substitution**\n\n**Plain English Explanation:**\nIf you’re working with a polynomial (like $x^2 + 5x - 3$), and you want to find its limit as $x$ approaches a certain number, you can just **plug that number in**. Polynomials are continuous everywhere, so they don’t have holes, jumps, or asymptotes.\n\n**Example:**\n\n$$\n\\lim_{x \\to 2}(x^2 + 3x - 1) = 2^2 + 3(2) - 1 = 4 + 6 - 1 = 9\n$$\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-3\"\u003e\u003c/a\u003e\n### **Theorem 3**\n\n**Limits of rational functions can be found by substitution if the limit of the denominator is not zero**\n\n**Plain English Explanation:**\nRational functions are fractions made of polynomials. If plugging in a number for $x$ **doesn't make the denominator zero**, then you can just plug the number in to find the limit.\n\n**Example (safe):**\n\n$$\n\\lim_{x \\to 1} \\frac{x^2 + 1}{x + 2} = \\frac{1^2 + 1}{1 + 2} = \\frac{2}{3}\n$$\n\nBut if the **denominator becomes zero**, substitution doesn’t work — you’ll need to try simplifying first (see below).\n\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"how-to-eliminate-zero-denominators-algebraically\"\u003e\u003c/a\u003e\n### **How to Eliminate Zero Denominators Algebraically**\n\nSometimes when you substitute, you get **0 in the denominator**, which is undefined. If this happens but the limit still exists, you can try:\n\n* **Canceling a common factor:**\n  Factor both top and bottom and cancel terms that are the same.\n\n  **Example:**\n\n  $$\n  \\lim_{x \\to 3} \\frac{x^2 - 9}{x - 3} = \\lim_{x \\to 3} \\frac{(x - 3)(x + 3)}{x - 3} = \\lim_{x \\to 3} (x + 3) = 6\n  $$\n\n* **Creating and canceling a common factor:**\n  Multiply top and bottom by a conjugate or do algebra to reveal a common factor that cancels.\n\n  **Example (with conjugate):**\n\n  $$\n  \\lim_{x \\to 0} \\frac{\\sqrt{x + 1} - 1}{x} \\\\\n  \\text{Multiply by the conjugate: } \\frac{\\sqrt{x + 1} - 1}{x} \\cdot \\frac{\\sqrt{x + 1} + 1}{\\sqrt{x + 1} + 1} = \\frac{x}{x(\\sqrt{x + 1} + 1)} = \\frac{1}{\\sqrt{x + 1} + 1}\n  $$\n\n  Now you can plug in $x = 0$:\n  $\\frac{1}{2}$\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-4\"\u003e\u003c/a\u003e\n### **Theorem 4**\n\n**The Sandwich Theorem (Squeeze Theorem)**\n\n**Plain English Explanation:**\nIf a function is \"trapped\" between two other functions that both approach the same limit, then the trapped function has to go to that same limit too.\n\n\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"extensions-of-the-limit-concept\"\u003e\u003c/a\u003e\n### **Extensions of the Limit Concept**\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"one-sided-limits\"\u003e\u003c/a\u003e\n#### **One-Sided Limits**\n\n**Plain English Explanation:**\nA **one-sided limit** looks at how a function behaves as the input gets close to a number **from just one side**:\n\n* **Left-hand limit**: what $f(x)$ approaches as $x$ comes **from the left** (written as $\\lim_{x \\to a^-} f(x)$)\n* **Right-hand limit**: what $f(x)$ approaches as $x$ comes **from the right** (written as $\\lim_{x \\to a^+} f(x)$)\n\n**Why it matters:**\nSometimes a function behaves **differently on each side** of a number. One-sided limits help us describe and analyze that.\n\n**Example:**\nFor the function\n\n$$\nf(x) = \n\\begin{cases}\n1, \u0026 x \u003c 0 \\\\\n2, \u0026 x \u003e 0\n\\end{cases}\n$$\n\n* $\\lim_{x \\to 0^-} f(x) = 1$\n* $\\lim_{x \\to 0^+} f(x) = 2$\n* Since the left and right limits **don’t match**, the full (two-sided) limit at $x = 0$ **does not exist**.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"infinite-limits\"\u003e\u003c/a\u003e\n#### **Infinite Limits**\n\n**Plain English Explanation:**\nAn **infinite limit** means that as $x$ gets close to some value, the function's output **grows without bound**—either positively or negatively.\n\n* If $f(x)$ becomes very large, we write:\n  $\\lim_{x \\to a} f(x) = \\infty$\n* If $f(x)$ becomes very negative:\n  $\\lim_{x \\to a} f(x) = -\\infty$\n\n**Important note:**\nInfinite limits **do not exist in the usual sense**, but we use the infinity symbol to describe their behavior.\n\n**Example:**\n\n$$\n\\lim_{x \\to 0^+} \\frac{1}{x} = \\infty  \n\\quad \\text{and} \\quad  \n\\lim_{x \\to 0^-} \\frac{1}{x} = -\\infty\n$$\n\nThe function blows up as it nears 0 from either side.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-5-relationship-between-one-sided-and-two-sided-limits\"\u003e\u003c/a\u003e\n### **Theorem 5: Relationship Between One-Sided and Two-Sided Limits**\n\n**Formal Version:**\nA two-sided limit exists **if and only if** both the left-hand and right-hand limits exist **and are equal**.\n\n$$\n\\lim_{x \\to a} f(x) = L  \n\\quad \\text{if and only if} \\quad  \n\\lim_{x \\to a^-} f(x) = L = \\lim_{x \\to a^+} f(x)\n$$\n\n---\n\n**Plain English Explanation:**\n\nTo say the limit of a function exists at a point, you **must** check from both sides:\n\n* If both sides approach the **same number**, the full limit exists and equals that number.\n* If they approach **different numbers** (or one side doesn’t exist), then the full limit **does not exist**.\n\n**Example:**\nUsing the same piecewise function as earlier:\n\n$$\nf(x) = \n\\begin{cases}\n1, \u0026 x \u003c 0 \\\\\n2, \u0026 x \u003e 0\n\\end{cases}\n$$\n\n* Left-hand limit = 1\n* Right-hand limit = 2\n  → Two-sided limit **does not exist**\n\n\n\n\u003c!-- TOC --\u003e\u003ca name=\"continuity\"\u003e\u003c/a\u003e\n## **Continuity**\n\n\n\n\u003c!-- TOC --\u003e\u003ca name=\"basic-terms\"\u003e\u003c/a\u003e\n### **Basic Terms**\n\n* A function is **continuous** if you can draw its graph **without lifting your pencil**.\n* More formally, a function is continuous at a point if the limit at that point **exists**, the function is **defined** there, and the two are **equal**.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"continuity-at-a-point\"\u003e\u003c/a\u003e\n### **Continuity at a Point**\n\nFor a function to be **continuous at $x = a$**, three things must be true:\n\n1. $f(a)$ is defined.\n2. $\\lim_{x \\to a} f(x)$ exists.\n3. $\\lim_{x \\to a} f(x) = f(a)$\n\nIf any of these are missing, the function is **not** continuous at that point.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"removable-discontinuity\"\u003e\u003c/a\u003e\n#### **Removable Discontinuity**\n\n* A \"hole\" in the graph.\n* The limit exists, but the function is either not defined there or is defined to the wrong value.\n\n**Example:**\n\n$$\nf(x) = \\frac{x^2 - 1}{x - 1}\n$$\n\nThis simplifies to $f(x) = x + 1$ for $x \\neq 1$, but there's a **hole at $x = 1$** because the original function was undefined there.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"jump-discontinuity\"\u003e\u003c/a\u003e\n#### **Jump Discontinuity**\n\n* The left and right limits exist, but they are **not equal**.\n* You see a \"jump\" in the graph.\n\n**Example:**\nPiecewise functions that switch values at a certain point often have jump discontinuities.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"infinite-discontinuity\"\u003e\u003c/a\u003e\n#### **Infinite Discontinuity**\n\n* The function heads toward **infinity** at the point.\n* Often seen with vertical asymptotes.\n\n**Example:**\n\n$$\nf(x) = \\frac{1}{x - 2}\n$$\n\nAs $x \\to 2$, the function becomes infinitely large or small.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"continuous-at-a-left-or-right-endpoint\"\u003e\u003c/a\u003e\n#### **Continuous at a Left or Right Endpoint**\n\nIf a function is only defined from one side (like an interval that starts at a certain point), we can only check **one-sided continuity**.\n\n* **Left endpoint:** use the **right-hand limit**.\n* **Right endpoint:** use the **left-hand limit**.\n\n**If the one-sided limit equals the function value, it’s continuous at that endpoint.**\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"continuity-on-intervals\"\u003e\u003c/a\u003e\n#### **Continuity on Intervals**\n\nA function is continuous on an interval if it is continuous **at every point** in that interval.\n\n* Open interval $(a, b)$: function must be continuous between the two ends.\n* Closed interval $[a, b]$: function must also be continuous **at the endpoints**, using one-sided limits.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"key-theorems-about-continuity\"\u003e\u003c/a\u003e\n## **Key Theorems About Continuity**\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-6-continuity-of-algebraic-combinations\"\u003e\u003c/a\u003e\n### **Theorem 6: Continuity of Algebraic Combinations**\n\n**Plain English Explanation:**\nIf two functions are continuous at a point, then you can:\n\n* **Add**, **subtract**, **multiply**, or **divide** them (except dividing by 0),\n* And the result will also be continuous at that point.\n\n**Example:**\nIf $f(x)$ and $g(x)$ are continuous at $x = 2$, then so is $f(x) + g(x)$, $f(x)g(x)$, etc.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-7-continuity-of-polynomials-and-rational-functions\"\u003e\u003c/a\u003e\n### **Theorem 7: Continuity of Polynomials and Rational Functions**\n\n**Plain English Explanation:**\n\n* **Polynomials** (like $x^2 + 3x - 1$) are continuous **everywhere**.\n* **Rational functions** (fractions made of polynomials) are continuous **wherever the denominator isn’t zero**.\n\n**Example:**\n\n$$\nf(x) = \\frac{x^2 - 1}{x - 2}\n$$\n\nThis is continuous everywhere **except** $x = 2$, where it’s undefined.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-8-continuity-of-composite-functions\"\u003e\u003c/a\u003e\n### **Theorem 8: Continuity of Composite Functions**\n\n**Plain English Explanation:**\n\nIf:\n\n* $g(x)$ is continuous at $x = a$,\n* $f(x)$ is continuous at $g(a)$,\n\nThen the composition $f(g(x))$ is **also continuous at $x = a$**.\n\n**Example:**\nIf $f(x) = \\sin x$ and $g(x) = x^2$, then $f(g(x)) = \\sin(x^2)$ is continuous because both parts are continuous.\n\n---\n\n\u003c!-- TOC --\u003e\u003ca name=\"theorem-9-the-intermediate-value-theorem-ivt\"\u003e\u003c/a\u003e\n\n### **Theorem 9: The Intermediate Value Theorem (IVT)**\n\n**Plain English Explanation:**\n\nIf a function is continuous on a closed interval $[a, b]$, and the function takes on values $f(a)$ and $f(b)$, then it must **hit every value between $f(a)$ and $f(b)$** somewhere in that interval.\n\nThis means the function **can’t skip over** any value—it has to pass through all of them.\n\n**Why it matters:**\nThe IVT is used to **prove that equations have solutions**. If $f(a)$ is negative and $f(b)$ is positive, there must be some point where $f(x) = 0$.\n\n**Example:**\nIf $f(1) = -3$ and $f(4) = 5$, and $f$ is continuous, then **somewhere between 1 and 4**, the function equals 0.\n\n---\n\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fp1xt%2Fmarkdown-notes-calculus","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fp1xt%2Fmarkdown-notes-calculus","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fp1xt%2Fmarkdown-notes-calculus/lists"}