A real function f:A, where A is said to be lower semi-continuous in x0 if


and f is said to be upper semi-continuous if



A real function is continuousMathworldPlanetmath in x0 if and only if it is both upper and lower semicontinuous in x0.

We can generalize the definition to arbitrary topological spacesMathworldPlanetmath as follows.

Let A be a topological space. f:A is lower semicontinuous at x0 if, for each ε>0 there is a neighborhoodMathworldPlanetmathPlanetmath U of x0 such that xU implies f(x)>f(x0)-ε.


Let f:[a,b] be a lower (upper) semi-continuous function. Then f has a minimum (maximum) in [a,b].

Title semi-continuous
Canonical name Semicontinuous
Date of creation 2013-03-22 12:45:41
Last modified on 2013-03-22 12:45:41
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Definition
Classification msc 26A15
Classification msc 54-XX
Synonym semicontinuous