Editing Extreme value theorem (section)

==Extension to semi-continuous functions==

If the continuity of the function ''f'' is weakened to [[semi-continuity]],
then the corresponding half of the boundedness theorem and the extreme value theorem hold and the values –∞ or +∞, respectively, from the [[extended real number line]] can be allowed as possible values.{{clarify|reason=It is not clear what is meant by " –∞ or +∞, respectively...can be allowed as possible values." What is the "respectively" in respect ''to''?|date=August 2024}}

A function <math>f : [a, b] \to [-\infty, \infty)</math> is said to be ''upper semi-continuous'' if <math display="block">\limsup_{y\to x} f(y) \le f(x) \quad \forall x \in [a, b].</math> 

{{Math theorem
|If a function {{math|''f'' : [''a'', ''b''] → {{closed-open|–∞, ∞}}}} is upper semi-continuous, then ''f'' is bounded above and attains its supremum.}}
{{Math proof
|proof=If <math>f(x) = - \infty</math> for all ''x'' in [''a'',''b''], then the supremum is also <math>-\infty</math> and the theorem is true. In all other cases, the proof is a slight modification of the proofs given above. In the proof of the boundedness theorem, the  upper semi-continuity of ''f'' at ''x'' only implies that the [[limit superior]] of the subsequence {''f''(''x<sub>n<sub>k</sub></sub>'')} is bounded above by ''f''(''x'') < ∞, but that is enough to obtain the contradiction. In the proof of the extreme value theorem, upper semi-continuity of ''f'' at ''d'' implies that the limit superior of the subsequence {''f''(''d<sub>n<sub>k</sub></sub>'')} is bounded above by ''f''(''d''), but this suffices to conclude that ''f''(''d'') = ''M''.&nbsp;[[Q.E.D.|∎]]
}}


Applying this result to &minus;''f'' proves a similar result for the infimums of lower semicontinuous functions. 
A function <math>f : [a, b] \to [-\infty, \infty)</math> is said to be ''lower semi-continuous'' if <math display="block">\liminf_{y\to x} f(y) \geq f(x)\quad \forall x \in [a, b].</math> 

{{Math theorem
|If a function {{math|''f'' : [''a'', ''b''] → {{open-closed|–∞, ∞}}}} is lower semi-continuous, then ''f'' is bounded below and attains its [[infimum]].
}}

A real-valued function is upper as well as lower semi-continuous, if and only if it is continuous in the usual sense. Hence these two theorems imply the boundedness theorem and the extreme value theorem.