Jump discontinuity

Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$ be a limit point of both $D \cap (-\infty, a)$ and $D \cap (a, \infty)$. The point $a$ is a jump discontinuity of $f$ if both one-sided limits $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ exist and are different.

Remarks

The difference of the two one-sided limits is the size of the jump. For example, the sign function has a jump discontinuity at $0$, of size $2$. Monotone functions have only jump discontinuities.

Explanations

L1 human

At a jump discontinuity, the graph approaches one height from the left and another height from the right. No choice of the value at $a$ can repair this.