Jump Discontinuity

High School

Definition

Also known as step discontinuity, this refers to a discontinuity that happens when a graph jumps from a connect section of a graph to another. It is also known as a discontinuity whose limits from both the left and right exists but don't equal one another.

Worked examples

\(\lim_{x \to 2^-} f(x) = 5\) and \(\lim_{x \to 2^+} f(x) = 3\)

The left and right limits exist but differ, so the function jumps from height 5 to height 3 at \(x = 2\).

\(f(x) = \begin{cases} x + 1 & x < 0 \)
\( x + 3 & x \ge 0 \end{cases}\) at \(x = 0\)

The graph jumps from 1 (approaching from left) to 3 (at and right of zero) — a jump discontinuity of size 2.

Common mistakes

If \(f(c)\) is defined, there is no jump discontinuity at \(c\) → A jump discontinuity exists when \(\lim_{x \to c^-} f(x) \ne \lim_{x \to c^+} f(x)\), regardless of \(f(c)\) The function value at the point does not prevent a jump; the mismatch of one-sided limits causes it.
All discontinuities are jump discontinuities → Removable and infinite discontinuities also exist A removable discontinuity has equal one-sided limits; an infinite discontinuity has limits of ±∞.
The size of the jump equals \(|f(c^-) - f(c^+)|\) → The jump size is \(|\lim_{x \to c^-} f(x) - \lim_{x \to c^+} f(x)|\) Use the one-sided limits, not the function values, since \(f(c)\) might be undefined or different.

Where you'll use it next

Jump discontinuities reappear in piecewise function analysis, Fourier series, and any area where you model abrupt changes — signals, step functions in differential equations, and continuity proofs in real analysis.

Found in 1 StudyPug lesson

Point of discontinuity

Grade 12 Math

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