In mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, semi-continuity (or semicontinuity) is a property of extended real-valued function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s that is weaker than continuity

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. An extended real-valued function f is upper (lower) semi-continuous at a point x₀ if, roughly speaking, the function values for arguments near x₀ are either close to f(x₀) or less than (greater than) f(x₀).
Consider the function f, piecewise

Piecewise

On mathematics, a piecewise-defined function is a function whose definition changes depending on the value of the independent variable...

 defined by f(x) = –1 for x < 0 and f(x) = 1 for x ≥ 0.