In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function ${\displaystyle f}$ is upper (respectively, lower) semicontinuous at a point ${\displaystyle x_{0}}$ if, roughly speaking, the function values for arguments near ${\displaystyle x_{0}}$ are not much higher (respectively, lower) than ${\displaystyle f\left(x_{0}\right).}$ Briefly, a function on a domain ${\displaystyle X}$ is lower semi-continuous if its epigraph ${\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}}$ is closed in ${\displaystyle X\times \mathbb {R} }$, and upper semi-continuous if ${\displaystyle -f}$ is lower semi-continuous.

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 ${\displaystyle x_{0}}$ to ${\displaystyle f\left(x_{0}\right)+c}$ for some ${\displaystyle c>0}$, then the result is upper semicontinuous; if we decrease its value to ${\displaystyle f\left(x_{0}\right)-c}$ 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.

Assume throughout that ${\displaystyle X}$ is a topological space and ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is a function with values in the extended real numbers ${\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]}$.

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called upper semicontinuous at a point ${\displaystyle x_{0}\in X}$ if for every real ${\displaystyle y>f\left(x_{0}\right)}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ such that ${\displaystyle f(x)<y}$ for all ${\displaystyle x\in U}$. Equivalently, ${\displaystyle f}$ is upper semicontinuous at ${\displaystyle x_{0}}$ if and only if $${\displaystyle \limsup _{x\to x_{0}}f(x)\leq f(x_{0})}$$ where lim sup is the limit superior of the function ${\displaystyle f}$ at the point ${\displaystyle x_{0}}$, defined as $${\displaystyle \limsup _{x\to x_{0}}f(x)=\inf _{U\ni x_{0}}\sup _{x\in U}f(x)}$$ where the infimum is over all neighborhoods of the point ${\displaystyle x_{0}}$.

If ${\displaystyle X}$ is a metric space with distance function ${\displaystyle d}$ and ${\displaystyle f(x_{0})\in \mathbb {R} ,}$ this can also be restated using an ${\displaystyle \varepsilon }$-${\displaystyle \delta }$ formulation, similar to the definition of continuous function. Namely, for each ${\displaystyle \varepsilon >0}$ there is a ${\displaystyle \delta >0}$ such that ${\displaystyle f(x)<f(x_{0})+\varepsilon }$ whenever ${\displaystyle d(x,x_{0})<\delta .}$

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called upper semicontinuous if it satisfies any of the following equivalent conditions:

A function ${\displaystyle f:X\to {\overline {\mathbb {R} }}}$ is called lower semicontinuous at a point ${\displaystyle x_{0}\in X}$ if for every real ${\displaystyle y<f\left(x_{0}\right)}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ such that ${\displaystyle f(x)>y}$ for all ${\displaystyle x\in U}$. Equivalently, ${\displaystyle f}$ is lower semicontinuous at ${\displaystyle x_{0}}$ if and only if $${\displaystyle \liminf _{x\to x_{0}}f(x)\geq f(x_{0})}$$ where ${\displaystyle \liminf }$ is the limit inferior of the function ${\displaystyle f}$ at point ${\displaystyle x_{0}.}$