Continuous Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Continuous Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

A continuous function can be drawn without lifting the pencil — there are no sudden jumps, gaps, or points that shoot to infinity.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Continuity at x = a requires three things: f(a) is defined, \lim_{x\to a} f(x) exists, and the limit equals f(a).

Common stuck point: A function can be continuous everywhere except certain points.

Sense of Study hint: Check the three conditions at the suspicious point: is f(a) defined, does the limit exist, and does the limit equal f(a)?

Worked Examples

Example 1

easy
Show that f(x) = 3x^2 - 5x + 2 is continuous at x = 1 using the three-part definition of continuity.

Solution

  1. 1
    Part 1 — f(1) exists: f(1) = 3(1)-5(1)+2 = 0. ✓
  2. 2
    Part 2 — \lim_{x\to1} f(x) exists: since f is a polynomial, the limit equals the function value. \lim_{x\to1}(3x^2-5x+2) = 3-5+2 = 0. ✓
  3. 3
    Part 3 — Limit equals function value: \lim_{x\to1}f(x) = 0 = f(1). ✓ All three conditions hold, so f is continuous at x=1.

Answer

f is continuous at x = 1
Continuity requires three conditions: the function value exists, the limit exists, and they are equal. Polynomials satisfy all three at every point, making them everywhere continuous.

Example 2

hard
Find where f(x) = \dfrac{x^2 - 4}{x - 2} is discontinuous, classify the discontinuity, and determine if it can be removed.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Is g(x) = \dfrac{1}{x} continuous on (-\infty, \infty)? If not, identify where and why.

Example 2

medium
Apply the Intermediate Value Theorem to show that h(x) = x^3 - x - 1 has a root in the interval [1, 2].
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition