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 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.