Recall that the lower real numbers are more general than the real numbers in just the way needed to guarantee that any inhabited set of lower reals has a supremum (by including ∞\infty, and in constructive mathematics by additionally relaxing the requirements of locatedness). Consequently, the range of any lower-real-valued function with an inhabited domain must have a supremum.

The Extreme Value Theorem states that such a range must also have an infimum when certain conditions are met. Specifically, we move to the realm of topology, where the natural lower-real-valued functions are the lower semicontinuous ones. So long as the domain of a lower semicontinuous map is compact (and inhabited), says the EVT, the range has an infimum, and furthermore this infimum is attained, becoming a minimum.

Statements

We first discuss the statement

for continuous functions

and then the general case

for semicontinuous function.

For continuous functions

Proof

Since continuous images of compact spaces are compact the image f([a,b])⊂ℝf([a,b]) \subset \mathbb{R} is a compact subspace.

Suppose this image did not contain its maximum. Then {(−∞,x)} x∈f([a,b])\{(-\infty,x)\}_{x \in f([a,b])} were an open cover of the image, and hence, by its compactness, there would be a finite subcover, hence a finite set (x 1<x 2<⋯<x n)(x_1 \lt x_2 \lt \cdots \lt x_n) of points x i∈f([a,b])x_i \in f([a,b]), such that the union of the (−∞,x i)(-\infty,x_i) and hence the single set (−∞,x n)(-\infty, x_n) alone would cover the image. This were in contradiction to the assumption that x n∈f([a,b])x_n \in f([a,b]) and hence we have a proof by contradiction.

Similarly for the minimum.

Proof

Since continuous images of compact spaces are compact the image f([a,b])⊂ℝf([a,b]) \subset \mathbb{R} is a compact subspace.

By the Heine-Borel theorem the image, being compact, is a bounded closed subset, hence a finite union of bounded closed intervals and singleton subsets. By continuity of ff, this union cannot be disjoint, for if f([a,b])f([a,b]) were a disjoint union C 1⊔C 2C_1 \sqcup C_2 of closed inhabited subsets, then also the pre-image were a disjoint union of closed inhabited subsets f −1(C 1)⊔f −1(C 2)f^{-1}(C_1) \sqcup f^{-1}(C_2), contradicting the fact that the pre-image of the image is the connected interval [a,b][a,b].

For semicontinuous functions

If II is a compact space, and if ff is an upper semicontinuous function from II to the upper real numbers [−∞,∞[[{-\infty,\infty}[, then the range of ff is not only bounded above and not only has a finite supremum, but it actually has a maximum value (unless II is empty).

Similarly, and consequently (by replacing ff with −f-f), if ff is lower semicontinuous to the lower real numbers ]−∞,∞]]{-\infty,\infty}], then ranf\ran f is bounded below by a finite infimum which is its minimum value. Consequently, if ff is continuous to the real numbers ]−∞,∞[]{-\infty,\infty}[, then ranf\ran f is bounded and has both a maximum and a minimum.)

In constructive mathematics, this statement is correct in locale theory (in a sense that we should explain here!), but if we are speaking of pointwise functions defined on a real interval, then it fails in general. However, we have approximate versions; in particular, it is constructive that ranf\ran f is bounded and has (if real-valued and pointwise-continuous) a real infimum and supremum. However, the EVT interacts subtly with the fan theorem; the fan theorem is equivalent to the statement that whenever a pointwise-continuous real-valued function ff on [0,1][0,1] satisfies m<f<Mm \lt f \lt M, then m<inffm \lt \inf f and supf<M\sup f \lt M, which (once the existence of inff\inf f and supf\sup f is granted) may be viewed as a contrapositive form of the EVT. (In fact, the fan theorem is separately equivalent to the statement that every continuous function on [0,1][0,1] is uniformly continuous and to the statement that every uniformly continuous function on [0,1][0,1] has this contrapositive property.) But without the fan theorem, even this contrapositive form cannot be proved (in marked contrast to similar theorems such as the intermediate value theorem and the mean value theorem), and in fact it is false in Russian constructivism (give a counterexample?). (Does the full semicontinuous version follow from the fan theorem?)

continuous images of compact spaces are compact

References

Wikipedia, Extreme value theorem
Karin U. Katz, Mikhail G. Katz, Taras Kudryk, Toward a clarity of the extreme value theorem (arXiv:1404.5658)

Last revised on July 5, 2017 at 05:49:18. See the history of this page for a list of all contributions to it.