semi thesis

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).}

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.

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