A continuous function ƒ(x) on the closed interval [a,b] showing the absolute max (red) and the absolute min (blue).

In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed interval [a,b], then f must attain its maximum and minimum value, each at least once. That is, there exist numbers c and d in [a,b] such that: In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. ... For other uses, see Calculus (disambiguation). ... This article is about functions in mathematics. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... The largest and the smallest element of a set are called extreme values, or extreme records. ...

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A weaker version of this theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is bounded on that interval. That is, there exist real numbers m and M such that: In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ...

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The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.

The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a compact space to a subset of the real numbers attains its maximum and minimum. In calculus, Rolles theorem states that if a function f is continuous on a closed interval and differentiable on the open interval , and then there is some number c in the open interval such that . Intuitively, this means that if a smooth curve is equal at two points then... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Proving the theorems

We look at the proof for the upper bound and the maximum of f. By applying these results to the function –f, the existence of the lower bound and the result for the minimum of f follows. Also note that everything in the proof is done within the context of the real numbers. In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... Please refer to Real vs. ...

We first prove the boundedness theorem, which is a step in the proof of the extreme value theorem. The basic steps involved in the proof of the extreme value theorem are:

1. Prove the boundedness theorem.
2. Find a sequence so that its image converges to the supremum of f.
3. Show that there exists a subsequence that converges to a point in the domain.
4. Use continuity to show that the image of the subsequence converges to the supremum.

Look up image in Wiktionary, the free dictionary. ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics, a subsequence of some sequence is a new sequence which is formed from the original sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. ... In mathematics, the domain of a function is the set of all input values to the function. ...

Proof of the boundedness theorem

Suppose the function f is not bounded above on the interval [a,b]. Then, by the Archimedean property of the real numbers, for every natural number n, there exists an x_n in [a,b] such that f(x_n) > n. This defines a sequence {x_n}. Because [a,b] is bounded, the Bolzano-Weierstrass theorem implies that there exists a convergent subsequence {} of {x_n}. Denote its limit by x. As [a,b] is closed, it contains x. Because f is continuous at x, we know that {f()} converges to the real number f(x). But f() > n_k ≥ k for every k, which implies that {f()} diverges to infinity. Contradiction. Therefore, f is bounded above on [a,b]. ∎ In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... For other senses of this word, see sequence (disambiguation). ... The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ... For other uses, see Infinity (disambiguation). ... This article is about Latin phrase Q.E.D., as used in proofs. ...

Proof of the extreme value theorem

We will now show that the function f has a maximum in the interval [a,b]. By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a d in [a,b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M – 1/n is not an upper bound for f. Therefore, there exists d_n in [a,b] so that M – 1/n < f(d_n). This defines a sequence {d_n}. Since M is an upper bound for f, we have M – 1/n < f(d_n) ≤ M for all n. Therefore, the sequence {f(d_n)} converges to M. In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is...

The Bolzano-Weierstrass theorem tells us that there exists a subsequence {}, which converges to some d and, as [a,b] is closed, d is in [a,b]. Since f is continuous at d, the sequence {f()} converges to f(d). But {f()} is a subsequence of {f(d_n)} that converges to M, so M = f(d). Therefore, f attains its supremum M at d. ∎ The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. ... This article is about Latin phrase Q.E.D., as used in proofs. ...

Examples

The following examples show why the function domain needs to be closed and bounded.

1. f(x) = x defined over [0,∞) is not bounded from above.
2. f(x) = x/(1 + x) defined over [0,∞) is bounded but does not attain its least upper bound 1.
3. f(x) = 1/x defined over (0,1] is not bounded from above.
4. f(x) = 1 – x defined over (0,1] is bounded but never attains its least upper bound 1.

Defining f(0) = 0 in the last two examples shows that both theorems require continuity on [a,b].

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: In mathematical analysis, semi-continuity (or semicontinuity) is a property of real-valued functions that is weaker than continuity. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ...

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

for all x in [a,b], then f is bounded above and attains its supremum.

Proof: If f(x) = –∞ for all x in [a,b], then the supremum is also –∞ 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()} 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()} is bounded above by f(d), but this suffices to conclude that f(d) = M. ∎ In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ... This article is about Latin phrase Q.E.D., as used in proofs. ...

Applying this result to −f proves:

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

for all x in [a,b], then f is bounded below and attains its infimum. In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all other elements of the subset. ...

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.

Topological formulation

In general topology, the extreme value theorem follows from the general fact that compactness is preserved under continuity, and the fact that a subset of the real line is compact if and only if it is both closed and bounded. In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

External links

• A Proof for extreme value theorem at cut-the-knot
• Boundedness Theorem on PlanetMath
• Extreme Value Theorem on PlanetMath
• Extreme Value Theorem by Jacqueline Wandzura with additional contributions by Stephen Wandzura, The Wolfram Demonstrations Project.
• Eric W. Weisstein, Extreme ValueT heorem at MathWorld.