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(xnk)} 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(dnk)} 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.
Read more about this topic: Extreme Value Theorem
Famous quotes containing the words extension and/or functions:
“We are now a nation of people in daily contact with strangers. Thanks to mass transportation, school administrators and teachers often live many miles from the neighborhood schoolhouse. They are no longer in daily informal contact with parents, ministers, and other institution leaders . . . [and are] no longer a natural extension of parental authority.”
—James P. Comer (20th century)
“Let us stop being afraid. Of our own thoughts, our own minds. Of madness, our own or others’. Stop being afraid of the mind itself, its astonishing functions and fandangos, its complications and simplifications, the wonderful operation of its machinery—more wonderful because it is not machinery at all or predictable.”
—Kate Millett (b. 1934)