Types of Continuity and Discontinuity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Types of Continuity and Discontinuity.

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 x = a if \lim_{x \to a} f(x) = f(a). Discontinuities are classified as removable (limit exists but doesn't equal f(a)), jump (left and right limits exist but differ), or infinite (function blows up to \pm\infty).

Continuous means you can draw the graph without lifting your pen. A removable discontinuity is a single hole you could fill in. A jump discontinuity is a gap where the function leaps to a different value. An infinite discontinuity is where the function shoots off to infinity (a vertical asymptote).

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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 requires three things: (1) f(a) is defined, (2) \lim_{x \to a} f(x) exists, and (3) the limit equals f(a). Each type of discontinuity corresponds to which condition fails.

Common stuck point: Removable discontinuities are often 'hidden' by algebraic simplification. After canceling, the simplified function may look continuous, but the original function still has a hole at the canceled point.

Sense of Study hint: Check the three conditions one by one: is f(a) defined? Does the limit exist? Do they match? Whichever fails tells you the type.

Worked Examples

Example 1

easy
Classify the discontinuity of f(x) = \dfrac{x^2 - 4}{x - 2} at x = 2.

Solution

  1. 1
    At x=2: denominator = 0, so f(2) is undefined.
  2. 2
    Simplify (for x \neq 2): \frac{(x-2)(x+2)}{x-2} = x+2.
  3. 3
    Limit: \lim_{x\to 2}(x+2) = 4. The limit exists.
  4. 4
    Since the limit exists but f(2) is undefined, this is a removable discontinuity (hole at (2,4)).

Answer

Removable discontinuity at x = 2 (hole at the point (2, 4)).
A removable discontinuity occurs when the two-sided limit exists but doesn't equal the function value (or the function is undefined there). It can be 'removed' by defining f(2) = 4.

Example 2

medium
Determine whether the piecewise function f(x) = \begin{cases} x^2 & x < 1 \\ 3 - x & x \geq 1 \end{cases} is continuous at x = 1.

Practice Problems

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

Example 1

easy
Classify the discontinuity of g(x) = \frac{1}{(x-3)^2} at x = 3.

Example 2

medium
Find the value of c that makes h(x) = \begin{cases} cx + 1 & x \leq 2 \\ x^2 - 1 & x > 2 \end{cases} continuous at x = 2.
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Background Knowledge

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

limit