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 un (x), n = 1,2, …, are lower semicontinuous and ≥0, the sum
is lower semicontinuous. Semicontinuous functions are Baire functions of class 1.