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 f {\displaystyle f} is upper (respectively, lower) semicontinuous at a point x 0 {\displaystyle x_{0}} if, roughly speaking, the function values for arguments near x 0 {\displaystyle x_{0}} are not much higher (respectively, lower) than f ( x 0 ) . {\displaystyle f\left(x_{0}\right).} Briefly, a function on a domain X {\displaystyle X} is lower semi-continuous if its epigraph { ( x , t ) ∈ X × R : t ≥ f ( x ) } {\displaystyle \{(x,t)\in X\times \mathbb {R} :t\geq f(x)\}} is closed in X × R {\displaystyle X\times \mathbb {R} } , and upper semi-continuous if − f {\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 x 0 {\displaystyle x_{0}} to f ( x 0 ) + c {\displaystyle f\left(x_{0}\right)+c} for some c > 0 {\displaystyle c>0} , then the result is upper semicontinuous; if we decrease its value to f ( x 0 ) − 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.
Definitions Assume throughout that X {\displaystyle X} is a topological space and f : X → R ¯ {\displaystyle f:X\to {\overline {\mathbb {R} }}} is a function with values in the extended real numbers R ¯ = R ∪ { − ∞ , ∞ } = [ − ∞ , ∞ ] {\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}=[-\infty ,\infty ]} .
Upper semicontinuity A function f : X → R ¯ {\displaystyle f:X\to {\overline {\mathbb {R} }}} is called upper semicontinuous at a point x 0 ∈ X {\displaystyle x_{0}\in X} if for every real y > f ( x 0 ) {\displaystyle y>f\left(x_{0}\right)} there exists a neighborhood U {\displaystyle U} of x 0 {\displaystyle x_{0}} such that f ( x ) < y {\displaystyle f(x)<y} for all x ∈ U {\displaystyle x\in U} . Equivalently, f {\displaystyle f} is upper semicontinuous at x 0 {\displaystyle x_{0}} if and only if
lim sup x → x 0 f ( x ) ≤ 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 f {\displaystyle f} at the point x 0 {\displaystyle x_{0}} , defined as
lim sup x → x 0 f ( x ) = inf x 0 ∈ U sup x ∈ U f ( x ) {\displaystyle \limsup _{x\to x_{0}}f(x)=\inf _{x_{0}\in U}\sup _{x\in U}f(x)}
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