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Mathematics

Continuity & the Intermediate Value Theorem

Limits, Removable vs. Jump Discontinuities, and Root-Finding by IVT — A TLDR Primer

Continuity shows up on every calculus exam — and most students lose points not because the algebra is hard, but because they never got a clear picture of what continuity actually means or why anyone cares.

This TLDR primer cuts straight to what you need. It builds the intuitive "no-jump, no-hole, no-blowup" picture of continuity, then nails it down with the precise limit-based definition. It classifies the three main types of discontinuities — removable, jump, and infinite — so you can identify them instantly on a graph or in a formula. It catalogs which functions are continuous by nature and shows you the algebra rules for building continuous functions from simpler ones.

Then it turns to the **Intermediate Value Theorem**: what it says, what each hypothesis is doing, and — critically — what it does *not* promise. The final applied section walks through sign-change arguments to prove equations have solutions, and introduces the bisection method as the IVT applied over and over in practice.

This guide is short by design. While the textbook buries these ideas under pages of theory and filler examples, this primer strips everything to essentials and gets you oriented fast. If you are prepping for an AP calculus limits and continuity unit, reviewing before a midterm, or helping a student who is stuck on the IVT, this is the place to start.

If you want to understand it, not just memorize it — grab this guide and get to work.

What you'll learn
  • State the three-part epsilon-free definition of continuity at a point in terms of limits.
  • Classify discontinuities as removable, jump, or infinite, and recognize each from a graph or formula.
  • Identify which standard functions are continuous on their domains and use algebraic rules to extend that quickly.
  • Apply the Intermediate Value Theorem to prove a root or solution exists on a given interval.
  • Use bisection driven by the IVT to locate roots numerically to a desired accuracy.
  • Recognize common student traps: continuity vs. differentiability, IVT requiring a closed interval, and the difference between 'a root exists' and 'find the root'.
What's inside
  1. 1. What Continuity Really Means
    Builds the intuitive 'no-jump, no-hole, no-blowup' picture and then nails it down with the limit-based three-part definition.
  2. 2. How Continuity Fails: A Tour of Discontinuities
    Classifies the three main ways a function can be discontinuous with graphs, formulas, and examples students actually see on exams.
  3. 3. Which Functions Are Continuous, and Why
    Catalogs the standard continuous functions and uses the algebra-of-continuous-functions rules to handle anything built from them.
  4. 4. The Intermediate Value Theorem
    States the IVT carefully, explains why each hypothesis matters, and shows what it does and does not promise.
  5. 5. Using the IVT to Find Roots
    Walks through sign-change arguments to prove equations have solutions, then introduces bisection as IVT applied repeatedly.
  6. 6. Why It Matters and What Comes Next
    Connects continuity and the IVT to the Extreme Value Theorem, the Mean Value Theorem, and the broader role of existence theorems in calculus.
Published by Solid State Press
Continuity & the Intermediate Value Theorem cover
TLDR STUDY GUIDES

Continuity & the Intermediate Value Theorem

Limits, Removable vs. Jump Discontinuities, and Root-Finding by IVT — A TLDR Primer
Solid State Press

Continuity & the Intermediate Value Theorem

Limits, Removable vs. Jump Discontinuities, and Root-Finding by IVT — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 What Continuity Really Means
  2. 2 How Continuity Fails: A Tour of Discontinuities
  3. 3 Which Functions Are Continuous, and Why
  4. 4 The Intermediate Value Theorem
  5. 5 Using the IVT to Find Roots
  6. 6 Why It Matters and What Comes Next
Chapter 1

What Continuity Really Means

Pick up a pencil and draw any curve you like — without lifting the tip from the paper. Whatever you drew is continuous. The moment you lift the pencil, skip to a new spot, or punch a hole in the curve, you have a discontinuity. That image is worth keeping in mind, but calculus demands something more precise, because "pencil doesn't leave paper" breaks down the instant you try to use it in a proof.

The Informal Picture

Three things can go wrong with a function at a point $x = a$:

A hole. The function might be undefined at $a$, or defined there but at the wrong height — as if someone erased one dot and redrew it somewhere else. The curve approaches a single value from both sides, but the actual output either doesn't exist or doesn't match.

A jump. The curve arrives at $a$ from the left at one height and from the right at a completely different height, with no bridge between them.

A blowup. The function shoots toward $\pm\infty$ near $a$, so there is no finite value to land on.

Any of these three failures — hole, jump, blowup — breaks continuity at that point. A function is continuous at a point $x = a$ if none of these failures occur: the function is defined there, the curve approaches that same single value from both sides, and the function's actual output equals that value. Everything "matches up."

Tightening the Definition with Limits

To make the informal picture rigorous, we use limits. Recall that the limit of $f(x)$ as $x$ approaches $a$, written $\lim_{x \to a} f(x) = L$, means that the output $f(x)$ gets arbitrarily close to $L$ as $x$ gets close to (but not equal to) $a$. The limit asks where the function is headed, not where it is.

Two one-sided versions matter here. The left-hand limit $\lim_{x \to a^-} f(x)$ asks where $f(x)$ heads as $x$ approaches $a$ from values smaller than $a$. The right-hand limit $\lim_{x \to a^+} f(x)$ asks the same from values larger than $a$. The two-sided limit $\lim_{x \to a} f(x)$ exists exactly when both one-sided limits exist and are equal.

With that vocabulary in place, here is the definition every calculus course uses:

$f$ is continuous at $x = a$ if all three of the following hold:

About This Book

If you are staring down an AP Calculus exam and still fuzzy on limits and continuity, or you are a college freshman who nodded along in lecture but cannot quite explain what "continuous" actually means, this guide is for you. It also works for tutors who need a clean, fast refresher before a session.

This is a calculus continuity study guide built for high school and early college students who need the core ideas without the textbook sprawl. It covers continuity and the Intermediate Value Theorem explained from first principles, removable vs. jump discontinuity with practice-ready examples, and IVT root-finding as a calculus prep strategy — including how the IVT fits into the broader landscape of calculus existence theorems. Concise and short by design, with no filler.

Read straight through in order, since each section builds on the last. Work every example as you go, then use the problem set at the end for your AP Calculus limits and continuity review, or as intermediate value theorem exam help the night before a test.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

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