Definition Let $f$ be a real-valued function defined on $S\subseteq\mathbb{R}$. The left-hand one-sided limit at $a\in\mathbb{R}$ is defined to be the real number $L^{-}$ such that for every $\epsilon>0$ there exists a $\delta>0$ such that $|f(x)-L^{-}|<\epsilon$ whenever $0<a-x<\delta$.

Analogously, the right-hand one-sided limit at $a\in\mathbb{R}$ is the real number $L^{+}$ such that for every $\epsilon>0$ there exists a $\delta>0$ such that $|f(x)-L^{+}|<\epsilon$ whenever $0<x-a<\delta$.

Common notations for the one-sided limits are

Sometimes, left-handed limits are referred to as limits from below while right-handed limits are from above.

Theorem The ordinary limit of a function exists at a point if and only if both one-sided limits exist at this point and are equal (to the ordinary limit).

Example The Heaviside unit step function, sometimes colloquially referred to as the diving board function, defined by

has the simplest kind of discontinuity at $x=0$, a jump discontinuity. Its ordinary limit does not exist at this point, but the one-sided limits do exist, and are