--- title: "Continuity and the Intermediate Value Theorem" description: "Define continuity at a point as the limit agreeing with the value there, classify the ways that can fail, and use the intermediate value theorem to guarantee a root inside an interval." canonical: https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV source: https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Continuity and the Intermediate Value Theorem Define continuity at a point as the limit agreeing with the value there, classify the ways that can fail, and use the intermediate value theorem to guarantee a root inside an interval. Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV ## Ready when they can - State the three conditions continuity at a point requires - Classify a discontinuity as removable, jump or infinite - Use a sign change on a continuous function to guarantee a root ## Lesson: Curves with no breaks A function is continuous at a point when its graph runs through that point with no break. Three things must hold at once. First, f(c) is defined, so a dot exists there. Second, the limit as x approaches c exists, so both sides settle on one height. Third, that limit equals f(c), so the height the graph nears is the height of the dot. One sentence can pack all three: if the limit as x approaches 5 exists and equals f(5), the function is continuous at 5. Note what is missing: no derivative appears. A corner can be perfectly continuous yet have no slope there. **Example.** Breaks come in three shapes. A removable break is a single hole: (x squared - 9) / (x - 3) is missing at x = 3, yet its limit is 6, so defining f(3) = 6 would patch it. A jump break splits the sides: with f(x) = x + 1 below 2 and f(x) = 5 at 2 and above, the left side nears 3 while the right side sits at 5, and the graph leaps. An infinite break blows up: 1 / x near 0 grows without bound in size instead of settling, which draws a vertical asymptote. Continuity pays off in the Intermediate Value Theorem. If f is continuous on a closed interval, it takes every value between its endpoint values somewhere inside, since an unbroken curve cannot skip a height. Take f(x) = x cubed + x - 1 on [0, 1]. Polynomials are continuous everywhere, and f(0) = -1 while f(1) = 1, so zero sits between the ends and some c inside has f(c) = 0: a root is guaranteed. The same promise covers f(x) = x squared - 2 on [1, 2], where -1 and 2 force at least one root. The theorem proves a crossing exists without saying where it sits. **Tip.** The theorem demands continuity across the whole closed interval, so check that first. A sign change alone is not enough: if f jumps somewhere inside, the curve is broken and no root is promised, even with the ends on opposite sides of zero. Roots already sitting at the ends also end the story early. Whenever a question asks what licenses a root claim, point at both halves: continuity on the interval plus the sign change. **Recap.** Continuity means limit meets value, and on an unbroken interval a sign change guarantees a root. ## Practice 16 questions on this page, each with its working shown. ## Needs first - [Locating Roots and Iterative Methods](https://lightmysky.com/learn/mathematics/locating-roots-and-iterative-methods-mt_2COO6TWerM) - [The Limit Laws and Indeterminate Forms](https://lightmysky.com/learn/mathematics/the-limit-laws-and-indeterminate-forms-mt_rAJs-V4Jlx) ## Opens up - [The Mean Value Theorem](https://lightmysky.com/learn/mathematics/the-mean-value-theorem-mt_09iApYVelx) - [The Epsilon-Delta Definition of a Limit](https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk) - [Continuity and Uniform Continuity](https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz)