Intermediate value theorem

The intermediate value theorem says that if a function, [math]\displaystyle{ f }[/math], is continuous over a closed interval [math]\displaystyle{ [a,b] }[/math], and is equal to [math]\displaystyle{ f(a) }[/math] and [math]\displaystyle{ f(b) }[/math] at either end of the interval, for any number, c, between [math]\displaystyle{ f(a) }[/math] and [math]\displaystyle{ f(b) }[/math], we can find an [math]\displaystyle{ x }[/math] so that [math]\displaystyle{ f(x) = c }[/math].

This means that if a continuous function's sign changes in an interval, we can find a root of the function in that interval. For example, if [math]\displaystyle{ f(1) = -1 }[/math] and [math]\displaystyle{ f(2) = 2 }[/math], we can find an [math]\displaystyle{ x }[/math] in the interval [math]\displaystyle{ [1,2] }[/math] that is a root of this function, meaning that for this value of x, [math]\displaystyle{ f(x)=0 }[/math], if [math]\displaystyle{ f }[/math] is continuous. This corollary is called Bolzano's theorem.

Intermediate Value Theorem Media

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  Intermediate value theorem: Let f be a continuous function defined on [a,b] and let s be a number with f(a) < s < f(b). Then there exists some x between a and b such that f(x) = s.

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  The intermediate value theorem