# Extreme value theorem
The **extreme value theorem** states that a real-valued function that is continuous on a closed, bounded interval \([a,b]\) attains both an absolute maximum and an absolute minimum on that interval[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima). It is sometimes called the Weierstrass extreme value theorem, after Karl Weierstrass (1815–1897)[2](https://mathworld.wolfram.com/ExtremeValueTheorem.html). The theorem is the existence guarantee behind one-variable optimization: it ensures that the problem of maximizing or minimizing a continuous quantity over a closed, bounded range actually has a solution before any calculus is used to find it[3](https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_%28Sloughter%29/01%3A_Derivatives/1.10%3A_Optimization).
| Key fact | Detail |
|---|---|
| Statement | If \( f \) is continuous on the closed, bounded interval \([a,b]\), there is a point in \([a,b]\) where \( f \) has an absolute maximum and a point where it has an absolute minimum[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima) |
| Both hypotheses needed | If the interval is open, or the function has even one point of discontinuity, an absolute maximum or minimum may fail to exist[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima) |
| General form | A continuous real-valued function on a nonempty compact space is bounded and attains its supremum and infimum[4](https://androma.org/theorems/304) |
| Where extrema occur | On a closed, bounded interval, absolute extrema occur at endpoints or at critical points of the function[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima) |
| Standard proof | The continuous image of a compact set is compact, and compact subsets of \( \mathbb{R} \) are closed and bounded, hence contain their supremum and infimum[4](https://androma.org/theorems/304) |
| Practical role | The theorem yields a straightforward algorithm for optimization problems on closed, bounded intervals: check the critical points and the endpoints[3](https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_%28Sloughter%29/01%3A_Derivatives/1.10%3A_Optimization) |
## Statement and hypotheses
As stated in introductory calculus, the theorem says that a function \( f \) that is continuous on the closed and bounded interval \([a,b]\) must attain a global maximum value somewhere in \([a,b]\), as well as a global minimum there[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima). An absolute extremum is the largest or smallest value the function takes on the whole interval, and it may be positive, negative, or zero[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima).
Both adjectives in "closed, bounded interval" carry weight. If the interval \( I \) is open, or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over \( I \)[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima). A function defined on a bounded interval can fail to attain an extremum because a boundary point is excluded, or because a jump discontinuity lets a nearby value escape; among three functions defined on bounded intervals, only the one continuous on the closed interval is guaranteed to have both an absolute maximum and an absolute minimum[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima).
The hypotheses are sufficient, not necessary. A function may have both an absolute maximum and an absolute minimum, just one extremum, or neither; for example \( f(x) = 1/(x^2+1) \) over \( (-\infty, \infty) \) has an absolute maximum of 1 at \( x = 0 \) even though its domain is neither closed nor bounded[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima). Similarly, \( f(x) = x^2 + 4 \) over \( (-\infty, \infty) \) has an absolute minimum of 4 at \( x = 0 \)[5](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/04%3A_Applications_of_Derivatives/4.07%3A_Applied_Optimization_Problems). What the theorem removes is uncertainty: under its hypotheses, existence is guaranteed without inspecting the particular function.
## Why the theorem holds
The standard proof passes through compactness. In the topology of the real line, a set is compact if every open cover admits a finite subcover; by the Heine-Borel theorem, a subset of \( \mathbb{R} \) is compact if and only if it is both closed and bounded, so the interval \([a,b]\) is compact[6](https://ms.uky.edu/~ken/ma570/lectures/lecture2/compact.web). The key lemma is that the continuous image of a compact set is compact: if \( f : X \to Y \) is continuous and \( K \subseteq X \) is compact, then \( f(K) \) is compact[7](https://notes.amittai.studio/real-analysis/continuity/continuity-metric-spaces).
From there the argument is short. Since \( f([a,b]) \) is a compact subset of \( \mathbb{R} \), it is closed and bounded[6](https://ms.uky.edu/~ken/ma570/lectures/lecture2/compact.web). A bounded set of real numbers has a supremum and an infimum by the completeness property of the real numbers, and because the set is closed it contains those values; the supremum of \( f([a,b]) \) is then \( f(c) \) for some \( c \in [a,b] \), giving the absolute maximum, and the infimum gives the absolute minimum[6](https://ms.uky.edu/~ken/ma570/lectures/lecture2/compact.web).
Two proof styles are common in real analysis courses: one using open covers and one using sequences[8](https://a-warded.org/open-book/university-real-analysis/lesson-21). In the sequential version, compactness is expressed as sequential compactness, every sequence in \( K \) having a subsequence converging to a point of \( K \); one takes a sequence whose images approach the supremum, extracts a convergent subsequence, and uses continuity to show the limit point maps to the supremum itself[7](https://notes.amittai.studio/real-analysis/continuity/continuity-metric-spaces). The theorem is usually proved in a course on real analysis rather than in elementary calculus[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima).
## General form on compact spaces
The interval statement is the special case of a more general result. Let \( X \) be a nonempty compact topological space and \( f : X \to \mathbb{R} \) a continuous function; then \( f \) is bounded, and there exist \( x_{\max}, x_{\min} \in X \) with \( f(x_{\max}) = \sup_{x \in X} f(x) \) and \( f(x_{\min}) = \inf_{x \in X} f(x) \)[4](https://androma.org/theorems/304). The same statement holds for nonempty compact metric spaces, proved by sequential compactness[7](https://notes.amittai.studio/real-analysis/continuity/continuity-metric-spaces), and for nonempty compact subsets \( K \) of \( \mathbb{R}^n \) with continuous \( f : K \to \mathbb{R} \)[9](https://david-duan.me/assets/course-notes/Math247-1.pdf).
The generalization shows exactly which hypothesis does the work. The proof of the interval statement used only that \([a,b]\) is compact, not that it is an interval[7](https://notes.amittai.studio/real-analysis/continuity/continuity-metric-spaces). Compactness adds a global restriction on the domain to the local control given by continuity, and together they force the function to be bounded and to reach its largest and smallest values[8](https://a-warded.org/open-book/university-real-analysis/lesson-21). This is one of the reasons compactness is a central concept in analysis: many theorems, including the extreme value theorem, hold on compact domains and can fail without compactness[10](https://ncatlab.org/nlab/show/extreme%2Bvalue%2Btheorem).
## Use in optimization
The theorem converts an existence question into a finite search. If \( f \) is continuous on a closed, bounded interval \( I \), the absolute maximum and absolute minimum must occur at endpoints of \( I \) or at critical points of \( f \) in \( I \): an extremum at an interior point is a local extremum, and by Fermat's theorem the point there is a critical point[1](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima). Because of this extreme-value property, there is a straightforward algorithm for optimization problems involving continuous functions on closed and bounded intervals[3](https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_%28Sloughter%29/01%3A_Derivatives/1.10%3A_Optimization).
This guarantee is what makes the standard calculus procedure sound. In applied problems such as minimizing production costs, minimizing packaging material for a fixed volume, or maximizing area, a quantity is modeled as a continuous function on a closed, bounded interval, and the extreme value theorem certifies that a maximum and a minimum exist before critical points and endpoints are compared[5](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/04%3A_Applications_of_Derivatives/4.07%3A_Applied_Optimization_Problems). The theorem guarantees that a solution exists without telling how to find it; locating it is the job of the derivative tests[11](https://www.math.toronto.edu/courses/mat237y1/20189/notes/Chapter1/S1.4.html).
The same structure extends to several variables. For a continuous function \( z = f(x,y) \) on a closed, bounded set \( S \), the analogous theorem guarantees absolute extrema, and the one-variable result is cited as the foundation of the multivariable version[12](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch13.S8.html).
## Related results
The extreme value theorem is one of a family of continuity theorems on compact domains. Continuous functions on compact metric spaces are also uniformly continuous, and continuous images preserve connectedness, from which the intermediate value theorem follows[13](https://www.cs.lewisu.edu/~harsyram/RealAnalysisIIWorkbookSp2020.pdf). The extreme value theorem in turn serves as the principal ingredient in proofs of the mean value theorem in one-variable calculus[6](https://ms.uky.edu/~ken/ma570/lectures/lecture2/compact.web).
## References
1. [4.3 Maxima and Minima, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima)
2. [Extreme Value Theorem, Wolfram MathWorld](https://mathworld.wolfram.com/ExtremeValueTheorem.html)
3. [1.10 Optimization, Yet Another Calculus Text, Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_%28Sloughter%29/01%3A_Derivatives/1.10%3A_Optimization)
4. [Extreme Value Theorem for Compact Spaces, Androma](https://androma.org/theorems/304)
5. [4.7 Applied Optimization Problems, Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Calculus/Calculus_%28OpenStax%29/04%3A_Applications_of_Derivatives/4.07%3A_Applied_Optimization_Problems)
6. [Compact Sets and Continuous Functions, University of Kentucky lecture notes](https://ms.uky.edu/~ken/ma570/lectures/lecture2/compact.web)
7. [Continuity on Metric Spaces, Amittai study notes](https://notes.amittai.studio/real-analysis/continuity/continuity-metric-spaces)
8. [Continuity on Compact Sets and the Extreme Value Theorem, A-Warded](https://a-warded.org/open-book/university-real-analysis/lesson-21)
9. [Continuous Functions and Compactness, Math 247 course notes](https://david-duan.me/assets/course-notes/Math247-1.pdf)
10. [Extreme value theorem, nLab](https://ncatlab.org/nlab/show/extreme%2Bvalue%2Btheorem)
11. [S1.4 Compactness, MAT237, University of Toronto](https://www.math.toronto.edu/courses/mat237y1/20189/notes/Chapter1/S1.4.html)
12. [13.8 Extreme Values, APEX Calculus](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch13.S8.html)
13. [math 36100:real analysis ii lecture notes](https://www.cs.lewisu.edu/~harsyram/RealAnalysisIIWorkbookSp2020.pdf)
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