Semi-continuity

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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...

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