Semi-continuity

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}f is upper (respectively, lower) semicontinuous at a point {\displaystyle x_{0}}x_{0} if, roughly speaking, the function values for arguments near {\displaystyle x_{0}}x_{0} are not much higher (respectively, lower) than {\displaystyle f\left(x_{0}\right).}{\displaystyle f\left(x_{0}\right).}

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}}x_{0} to {\displaystyle f\left(x_{0}\right)+c}{\displaystyle f\left(x_{0}\right)+c} for some {\displaystyle c>0}c>0, then the result is upper semicontinuous; if we decrease its value to {\displaystyle f\left(x_{0}\right)-c}{\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.[1]

An upper semicontinuous function that is not lower semicontinuous. The solid blue dot indicates {\displaystyle f\left(x_{0}\right).}{\displaystyle f\left(x_{0}\right).}

A lower semicontinuous function that is not upper semicontinuous. The solid blue dot indicates {\displaystyle f\left(x_{0}\right).}{\displaystyle f\left(x_{0}\right).}

1 Definitions

Assume throughout that {\displaystyle X}X is a topological space and {\displaystyle f:X\to {\overline {\mathbb {R} }}}{\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 ]}{\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup {-\infty ,\infty }=[-\infty ,\infty ]}.

1.1 Upper semicontinuity

A function {\displaystyle f:X\to {\overline {\mathbb {R} }}}{\displaystyle f:X\to {\overline {\mathbb {R} }}} is called upper semicontinuous at a point {\displaystyle x_{0}\in X}{\displaystyle x_{0}\in X} if for every real {\displaystyle y>f\left(x_{0}\right)}{\displaystyle y>f\left(x_{0}\right)} there exists a neighborhood {\displaystyle U}U of {\displaystyle x_{0}}x_{0} such that {\displaystyle f(x)

{\displaystyle \limsup {x\to x{0}}f(x)\leq f(x_{0})}{\displaystyle \limsup {x\to x{0}}f(x)\leq f(x_{0})}
where lim sup is the limit superior of the function {\displaystyle f}f at the point {\displaystyle x_{0}}x_{0}.
A function {\displaystyle f:X\to {\overline {\mathbb {R} }}}{\displaystyle f:X\to {\overline {\mathbb {R} }}} is called upper semicontinuous if it satisfies any of the following equivalent conditions:[2]

(1) The function is upper semicontinuous at every point of its domain.
(2) All sets {\displaystyle f^{-1}([-\infty ,y))={x\in X:f(x) (3) All superlevel sets {\displaystyle {x\in X:f(x)\geq y}}{\displaystyle {x\in X:f(x)\geq y}} with {\displaystyle y\in \mathbb {R} }{\displaystyle y\in \mathbb {R} } are closed in {\displaystyle X}X.
(4) The hypograph {\displaystyle {(x,t)\in X\times \mathbb {R} :t\leq f(x)}}{\displaystyle {(x,t)\in X\times \mathbb {R} :t\leq f(x)}} is closed in {\displaystyle X\times \mathbb {R} }{\displaystyle X\times \mathbb {R} }.
(5) The function is continuous when the codomain {\displaystyle {\overline {\mathbb {R} }}}{\displaystyle {\overline {\mathbb {R} }}} is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals {\displaystyle [-\infty ,y)}{\displaystyle [-\infty ,y)}.

1.2 Lower semicontinuity

A function {\displaystyle f:X\to {\overline {\mathbb {R} }}}{\displaystyle f:X\to {\overline {\mathbb {R} }}} is called lower semicontinuous at a point {\displaystyle x_{0}\in X}x_{0}\in X if for every real {\displaystyle yy}{\displaystyle f(x)>y} for all {\displaystyle x\in U}x\in U. Equivalently, {\displaystyle f}f is lower semicontinuous at {\displaystyle x_{0}}x_{0} if and only if

{\displaystyle \liminf {x\to x{0}}f(x)\geq f(x_{0})}{\displaystyle \liminf {x\to x{0}}f(x)\geq f(x_{0})}
where {\displaystyle \liminf }\liminf is the limit inferior of the function {\displaystyle f}f at point {\displaystyle x_{0}}x_{0}.
A function {\displaystyle f:X\to {\overline {\mathbb {R} }}}{\displaystyle f:X\to {\overline {\mathbb {R} }}} is called lower semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is lower semicontinuous at every point of its domain.
(2) All sets {\displaystyle f^{-1}((y,\infty ])={x\in X:f(x)>y}}{\displaystyle f^{-1}((y,\infty ])={x\in X:f(x)>y}} with {\displaystyle y\in \mathbb {R} }{\displaystyle y\in \mathbb {R} } are open in {\displaystyle X}X, where {\displaystyle (y,\infty ]={t\in {\overline {\mathbb {R} }}:t>y}}{\displaystyle (y,\infty ]={t\in {\overline {\mathbb {R} }}:t>y}}.
(3) All sublevel sets {\displaystyle {x\in X:f(x)\leq y}}{\displaystyle {x\in X:f(x)\leq y}} with {\displaystyle y\in \mathbb {R} }{\displaystyle y\in \mathbb {R} } are closed in {\displaystyle X}X.
(4) The epigraph {\displaystyle {(x,t)\in X\times \mathbb {R} :t\geq f(x)}}{\displaystyle {(x,t)\in X\times \mathbb {R} :t\geq f(x)}} is closed in {\displaystyle X\times \mathbb {R} }{\displaystyle X\times \mathbb {R} }.
(5) The function is continuous when the codomain {\displaystyle {\overline {\mathbb {R} }}}{\displaystyle {\overline {\mathbb {R} }}} is given the right order topology. This is just a restatement of condition (2) since the right order topology is generated by all the intervals {\displaystyle (y,\infty ]}{\displaystyle (y,\infty ]}.

2 Examples

3 Properties

4 See also