one-sided limit

Definition Let f be a real-valued function defined on S. The left-hand one-sided limit at a is defined to be the real number L- such that for every ϵ>0 there exists a δ>0 such that |f(x)-L-|<ϵ whenever 0<a-x<δ.

Analogously, the right-hand one-sided limit at a is the real number L+ such that for every ϵ>0 there exists a δ>0 such that |f(x)-L+|<ϵ whenever 0<x-a<δ.

Common notations for the one-sided limits are

L+ = f(x+)=limxa+f(x)=limxaf(x),
L- = f(x-)=limxa-f(x)=limxaf(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)={0 if x<01 if x0

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

limx0-H(x)=0 and limx0+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