# Semicontinuous Function

## Semicontinuous Function

a concept in mathematical analysis. A lower semicontinuous function at a point *x*_{0} is a function such that

Correspondingly for an upper function,

In other words, a function is lower semicontinuous at *x*_{0} if for every ε > 0 a number δ > 0 can be found such that ǀ*x* — *x*_{0}ǀ < δ implies *f*(*x*_{0}) — *f*(*x*) < ε (not in absolute value!).

A function that is both upper and lower semicontinuous is continuous in the usual sense. A number of properties of semi-continuous functions are analogous to those of continuous functions. For example, if *f(x*) and *g(x*) are lower semicontinuous, their sum and product are also lower semicontinuous; likewise, a lower semicontinuous function on an interval attains its minimum. If the functions *u _{n}* (

*x*),

*n*= 1,2, …, are lower semicontinuous and ≥0, the sum

is lower semicontinuous. Semicontinuous functions are Baire functions of class 1.