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Intermediate Value Theorem

Intermediate Value Theorem states that a function f continuous in the closed interval [a,b] takes every value in between f(a) and f(b).

Theorem 1

Let f be a function defined on a closed interval [a, b]. Let c be a number lying between f(a) and f(b) (i.e. f(a) < c < f(b) if f(a) < f(b) or f(b) < c < f(a) if f(b) < f(a)). Then there exists at least one point xo ∈ [a, b] such that f(xo) = c.


The importance of this theorem is as follows : If we have a continuous function f defined on a closed interval [a, b], then the theorem guarantees the existence of a solution of the equation f(x) = c, where c is as in Theorem 1. However, it does not say what the solution is.


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