Jump Discontinuity — Definition, Formula & Examples

A jump discontinuity is a point where a function "jumps" from one value to another — the limit from the left and the limit from the right both exist, but they are different.

A function $f$ has a jump discontinuity at $x = c$ if both one-sided limits $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$ exist and are finite, but $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$ . Because the two-sided limit does not exist, $f$ is discontinuous at $c$ .

Key Formula

\lim_{x \to c^-} f(x) = L \neq R = \lim_{x \to c^+} f(x)

Where:

$c$ = The point where the potential discontinuity occurs
$L$ = The finite left-hand limit of f as x approaches c
$R$ = The finite right-hand limit of f as x approaches c

How It Works

To check for a jump discontinuity at

x = c

, compute the left-hand limit and the right-hand limit separately. If both limits are finite real numbers but differ, the function has a jump discontinuity there. The size of the jump is

\left|\lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x)\right|

. Piecewise-defined functions are the most common source of jump discontinuities.

Worked Example

Problem: Determine whether the piecewise function has a jump discontinuity at

x = 2

f(x) = \begin{cases} x + 1 & x < 2 \\ x + 4 & x \geq 2 \end{cases}

Step 1: Compute the left-hand limit as

x \to 2^-

using the piece

f(x) = x + 1

\lim_{x \to 2^-} f(x) = 2 + 1 = 3

Step 2: Compute the right-hand limit as

x \to 2^+

using the piece

f(x) = x + 4

\lim_{x \to 2^+} f(x) = 2 + 4 = 6

Step 3: Since both one-sided limits exist but

3 \neq 6

, the function has a jump discontinuity at

x = 2

with a jump size of

|6 - 3| = 3

Answer:

f

has a jump discontinuity at

x = 2

. The function jumps by 3 units.

Why It Matters

Jump discontinuities appear frequently in real-world models — tax brackets, shipping cost functions, and electrical signals all exhibit sudden jumps. Recognizing them is also essential when applying theorems like the Intermediate Value Theorem, which requires continuity on a closed interval.

Common Mistakes

Mistake: Confusing a jump discontinuity with an essential (infinite) discontinuity.

Correction: At a jump discontinuity, both one-sided limits are finite but unequal. If either one-sided limit is infinite or fails to exist entirely, the discontinuity is of a different type.