one-sided limit

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

	$\displaystyle L^{+}$	$\displaystyle=$	$\displaystyle f(x+)=\lim_{x\to a^{+}}f(x)=\lim_{x\searrow a}f(x),$
	$\displaystyle L^{-}$	$\displaystyle=$	$\displaystyle f(x-)=\lim_{x\to a^{-}}f(x)=\lim_{x\nearrow a}f(x).$

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

H(x)=\begin{cases}0&\mbox{ if }~{}x<0\\ 1&\mbox{ if }~{}x\geq 0\end{cases}

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

\lim_{x\to 0^{-}}H(x)=0\mbox{ and }\lim_{x\to 0^{+}}H(x)=1.

Title	one-sided limit
Canonical name	OnesidedLimit
Date of creation	2013-03-22 12:40:28
Last modified on	2013-03-22 12:40:28
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 26A06
Synonym	limit from below
Synonym	limit from above
Synonym	left-sided limit
Synonym	left-handed limit
Synonym	right-sided limit
Synonym	right-handed limit
Related topic	Limit
Related topic	OneSidedDerivatives
Related topic	IntegratingTanXOver0fracpi2
Related topic	OneSidedContinuity
Defines	Heaviside unit step function