Semicontinuous Function

Semicontinuous Function 

a concept in mathematical analysis. A lower semicontinuous function at a point x₀ is a function such that

Correspondingly for an upper function,

In other words, a function is lower semicontinuous at x₀ if for every ε > 0 a number δ > 0 can be found such that ǀx — x₀ǀ < δ implies f(x₀) — 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.