In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, semicontinuity (or semi-continuity) is a property of
extended real-valued
functions that is weaker than
continuity. An extended real-valued function
is upper (respectively, lower) semicontinuous at a point
if, roughly speaking, the function values for arguments near
are not much higher (respectively, lower) than
A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point
to
for some
, then the result is upper semicontinuous; if we decrease its value to
then the result is lower semicontinuous.

The notion of upper and lower semicontinuous function was first introduced and studied by
René Baire in his thesis in 1899.
Definitions
Assume throughout that
is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and
is a function with values in the
extended real numbers