Semi-continuity

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, semicontinuity (or semi-continuity) is a property of <a href="/facts/Extended_real_number/65Gs7QS0">extended real</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> that is weaker than <a href="/facts/Continuous_function/sKbl02pB">continuity</a>. 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 <a href="/facts/Epigraph_(mathematics)/qGYcHa8t">epigraph</a> 
 
 
 
 {
 (
 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 <a href="/facts/Ren%25C3%25A9_Baire/fdxa8oK5">René Baire</a> in his thesis in 1899.

Semi-continuity open-in-new

Semi-continuity