# What is jump discontinuity in math?

Define a function $f$. There exists a jump discontinuity at a point $a$ if ${\lim}_{x \to {a}^{-}} f \left(x\right) = \alpha$ and ${\lim}_{x \to {a}^{+}} f \left(x\right) = \beta$ such that $\alpha$ and $\beta$ are real numbers (excluding $\pm \infty$) and $\alpha \ne \beta$. An example is the signum function $s g n \left(x\right) = | x \frac{|}{x}$ at $x = 0$. Since ${\lim}_{x \rightarrow {0}^{-}} s g n \left(x\right) = - 1$ and ${\lim}_{x \rightarrow {0}^{+}} s g n \left(x\right) = 1$, there is a jump discontinuity at $x = 0$.