Definition Let $f$ be a real-valued function defined on $S \subseteq \sR$ . The left-hand one-sided limit at $a\in \sR$ 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 $a\in \sR$ 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 \begin{eqnarray*} L^+ &=& f(x+) = \lim_{x \to a^+} f(x) =\lim_{x \searrow a} f(x), \\ L^- &=&f(x-) = \lim_{x \to a^-} f(x) =\lim_{x \nearrow a} f(x). \end{eqnarray*}

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