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The extreme value theorem states that if a function f is defined on a closed interval [,] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists [,] with () for all [,].
Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.
The Radon–Nikodym theorem [14] states that if is absolutely continuous with respect to , and both measures are σ-finite, then has a density, or "Radon-Nikodym derivative", with respect to , which means that there exists a -measurable function taking values in [, +), denoted by = /, such that for any -measurable set we have: =.
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function: [,] can be represented as a superposition of continuous single-variable functions.
The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.
The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on are shown to suffice, as is detailed below.
Heine–Cantor theorem — If : is a continuous function between two metric spaces and , and is compact, then is uniformly continuous. An important special case of the Cantor theorem is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.