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Mastering Calculus: Discontinuous Functions Explained (With Examples)

In calculus, a function is considered continuous if you can trace its graph without ever lifting your pencil. But what happens when the math forces you to lift that pencil? When a function breaks, jumps, or shoots off toward infinity, it becomes a discontinuous function.

Understanding where and why a function breaks is just as important as knowing where it is smooth. To locate a discontinuity, we look for points where a function fails the strict three-part continuity test.

Here is exactly what causes a function to fracture, the distinct types of discontinuities you will encounter, and examples of how they look mathematically.

What Makes a Function Discontinuous?

A function f(x) is discontinuous at a point x = a if it fails at least one of the three following conditions:

  1. f(a) must be defined. (There cannot be a hole or an asymptote).
  2. The limit as x \to a must exist. (The left and right sides must approach the same numerical value).
  3. The limit must equal the function’s value. (\lim_{x \to a} f(x) = f(a)).

When a function fails this test, it generally falls into one of three major categories: Removable (a single missing point), Jump (a sudden vertical shift), or Infinite (a vertical asymptote).

5 Examples of Discontinuities in Action

Let’s break down the different ways a function can fail the continuity test and classify the resulting algebraic breaks.

Example 1: The Standard Hole (Removable Discontinuity)

Let’s test f(x) = \frac{x^2 - 4}{x - 2} at the point x = 2.

To check condition 1, we plug in 2:

f(2) = \frac{2^2 - 4}{2 - 2} = \frac{0}{0}
Because you cannot divide by zero, f(2) is undefined. The function immediately fails Condition 1.

If you were to evaluate the limit by factoring the numerator to (x-2)(x+2), the limit evaluates to 4. Because the limit exists but the point itself does not, this creates a single, infinitesimally small gap in the graph known as a removable discontinuity.

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Example 2: The Misplaced Point (Failed Condition 3)

Sometimes a function is defined, and the limit exists, but they do not match up. Consider a piecewise function where:
f(x) = \frac{x^2 - 9}{x - 3} when x \neq 3
f(x) = 10 when x = 3

Let’s test for continuity at x = 3:

  1. f(3) = 10. (Condition 1 passes).
  2. \lim_{x \to 3} f(x) evaluates to 6 (after factoring). (Condition 2 passes).
  3. Does \lim_{x \to 3} f(x) = f(3)?
    6 \neq 10

The function fails Condition 3. The graph looks like a normal line heading toward a height of 6, but the actual point has been plucked out and moved up to 10. This is another variation of a removable discontinuity.

Example 3: The Gap (Jump Discontinuity)

Jump discontinuities are most common in piecewise functions where the graph literally disconnects and resumes at a different height. Let’s look at a function at x = 0:
f(x) = -1 when x < 0
f(x) = 1 when x \ge 0

Test for continuity at x = 0:

  1. f(0) = 1. (Condition 1 passes).
  2. Evaluate the limit from the left and right.
    From the left (x < 0): \lim_{x \to 0^-} -1 = -1
    From the right (x > 0): \lim_{x \to 0^+} 1 = 1

Because the left-hand limit (-1) does not equal the right-hand limit (1), the overall limit does not exist. The function fails Condition 2, creating a jump discontinuity.

Example 4: The Vertical Asymptote (Infinite Discontinuity)

Rational functions often feature denominators that completely blow up. Let’s look at f(x) = \frac{1}{(x - 1)^2} at the point x = 1.

If you plug in 1, you get \frac{1}{0}, which is undefined, immediately failing Condition 1.

If you check the limits, as x approaches 1 from either the left or the right, the denominator gets incredibly small, driving the function’s value toward positive infinity. Because the graph shoots infinitely upward and never meets at a real number, the limit does not exist. This is known as an infinite discontinuity.

Example 5: The Wild Oscillation (Essential Discontinuity)

There is a rare but mathematically important type of discontinuity that doesn’t hole, jump, or shoot to infinity. Instead, it behaves erratically.

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f(x) = \sin\left(\frac{1}{x}\right)
Test this at x = 0. Plugging in 0 results in division by zero, so f(0) is undefined (fails Condition 1).

If you try to find the limit as x approaches 0, the fraction \frac{1}{x} approaches infinity. The sine function responds by oscillating endlessly between -1 and 1 at an increasingly rapid pace. Because it never settles on a single number, the limit does not exist. This extreme behavior is classified as an essential discontinuity.

Discontinuities define the boundaries of calculus. Before you can calculate the slope of a curve or the area beneath it, you must know your function’s domain and limitations. Identifying where a function breaks down via holes, jumps, or asymptotes ensures you don’t accidentally try to apply derivatives or integrals across mathematical barriers that do not exist.

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