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Discontinuity — Definition, Types & Examples

Discontinuity

A point at which the graph of a relation or function is not connected. Discontinuities can be classified as either removable or essential. There are several kinds of essential discontinuities, one of which is the step discontinuity.

 

Two graphs on x-y axes showing discontinuities: left has a parabola with open circle (removable discontinuity); right shows a...

Key Formula

A function f is continuous at x=c if and only if:\text{A function } f \text{ is continuous at } x = c \text{ if and only if:} lim⁡x→cf(x)=f(c)\lim_{x \to c} f(x) = f(c) A discontinuity occurs at x=c when this condition fails.\text{A discontinuity occurs at } x = c \text{ when this condition fails.}
Where:
  • ff = The function being analyzed
  • cc = The x-value where continuity is being tested
  • lim⁡x→cf(x)\lim_{x \to c} f(x) = The limit of f(x) as x approaches c
  • f(c)f(c) = The actual value of the function at x = c

Worked Example

Problem: Determine whether the function f(x) = (x² − 4)/(x − 2) has a discontinuity, and if so, classify it.
Step 1: Check the domain. The denominator is zero when x = 2, so f(2) is undefined. This is a candidate for a discontinuity.
x−2=0  ⟹  x=2x - 2 = 0 \implies x = 2
Step 2: Factor the numerator and simplify the expression to see what happens near x = 2.
f(x)=x2−4x−2=(x−2)(x+2)x−2=x+2(x≠2)f(x) = \frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2 \quad (x \neq 2)
Step 3: Compute the limit as x approaches 2.
lim⁡x→2f(x)=lim⁡x→2(x+2)=4\lim_{x \to 2} f(x) = \lim_{x \to 2}(x + 2) = 4
Step 4: The limit exists and equals 4, but f(2) is undefined. Since the condition for continuity fails only because f(c) is missing (or could be redefined to equal the limit), this is a removable discontinuity — a "hole" in the graph at (2, 4).
lim⁡x→2f(x)=4≠f(2)  (undefined)\lim_{x \to 2} f(x) = 4 \neq f(2) \;(\text{undefined})
Answer: There is a removable discontinuity (hole) at x = 2. The graph looks like the line y = x + 2 with a single missing point at (2, 4).

Another Example

This example shows an essential discontinuity (jump type), unlike the first example which was removable. Here the limit itself fails to exist because the one-sided limits disagree.

Problem: Identify and classify any discontinuities of the piecewise function: f(x) = 1 for x < 0, and f(x) = 3 for x ≥ 0.
Step 1: Check the value of f at the boundary point x = 0. By the definition of the piecewise function, f(0) = 3.
f(0)=3f(0) = 3
Step 2: Compute the left-hand limit as x approaches 0 from the negative side.
lim⁡x→0−f(x)=1\lim_{x \to 0^-} f(x) = 1
Step 3: Compute the right-hand limit as x approaches 0 from the positive side.
lim⁡x→0+f(x)=3\lim_{x \to 0^+} f(x) = 3
Step 4: Since the left-hand and right-hand limits are not equal, the two-sided limit does not exist. This means the discontinuity cannot be removed by changing a single point — it is an essential discontinuity, specifically a jump (step) discontinuity.
lim⁡x→0−f(x)=1≠3=lim⁡x→0+f(x)\lim_{x \to 0^-} f(x) = 1 \neq 3 = \lim_{x \to 0^+} f(x)
Answer: There is a step (jump) discontinuity at x = 0. The graph jumps from y = 1 to y = 3.

Frequently Asked Questions

What is the difference between a removable and essential discontinuity?
A removable discontinuity occurs when the limit exists at the point but the function is either undefined or defined to a different value — you can "fix" it by redefining the function at that single point. An essential (non-removable) discontinuity occurs when the limit itself does not exist, such as in a jump, infinite asymptote, or oscillating behavior. No single-point redefinition can repair it.
How do you find discontinuities of a function?
First, identify points where the function might break: where the denominator equals zero, where a piecewise definition changes, or where the domain has a boundary. Then test the continuity condition at each candidate point: check that f(c) is defined, the limit as x approaches c exists, and the limit equals f(c). Any failure means a discontinuity at that point.
Can a function have infinitely many discontinuities?
Yes. For example, the floor function f(x) = ⌊x⌋ has a jump discontinuity at every integer. Some functions, like the Dirichlet function (which equals 1 on rationals and 0 on irrationals), are discontinuous at every real number.

Removable Discontinuity vs. Essential Discontinuity

Removable DiscontinuityEssential Discontinuity
DefinitionThe limit exists at x = c, but f(c) is undefined or not equal to the limitThe two-sided limit does not exist at x = c
Graph appearanceA "hole" — a single missing or misplaced pointA jump, vertical asymptote, or wild oscillation
Can it be fixed?Yes — redefine f(c) to equal the limitNo — cannot be repaired by changing one point
Common examplef(x) = (x² − 1)/(x − 1) at x = 1f(x) = 1/x at x = 0 (infinite); floor function at integers (jump)

Why It Matters

Discontinuities appear throughout precalculus and calculus whenever you study limits, derivatives, or integrals. Many key theorems — such as the Intermediate Value Theorem and the Fundamental Theorem of Calculus — require continuity, so you must identify discontinuities to know when those results apply. In real-world modeling, discontinuities represent sudden changes like tax brackets, phase transitions, or switching behavior in circuits.

Common Mistakes

Mistake: Assuming that if a function is undefined at a point, it automatically has a non-removable discontinuity there.
Correction: An undefined point may be a removable discontinuity if the limit exists. Always compute the limit before classifying. For example, (x² − 9)/(x − 3) is undefined at x = 3, but the limit is 6, making it removable.
Mistake: Forgetting to check one-sided limits separately at piecewise boundaries.
Correction: When a function is defined by different formulas on each side of a point, you must verify that the left-hand limit and right-hand limit are equal. If they differ, you have a jump discontinuity, even if the function is defined at that point.

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