Extreme Value Theorem - Extension To Semi-continuous Functions

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. More precisely:

Theorem: If a function f : → [–∞,∞) is upper semi-continuous, meaning that

for all x in, then f is bounded above and attains its supremum.

Proof: If f(x) = –∞ for all x in, then the supremum is also –∞ and the theorem is true. In all other cases, the proof is a slight modification of the proofs given below. 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_nk)} 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_nk)} is bounded above by f(d), but this suffices to conclude that f(d) = M. ∎

Applying this result to −f proves:

Theorem: If a function f : → (–∞,∞] is lower semi-continuous, meaning that

for all x in, 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.