# Monotonic functions

Give an example to show that the intermediate value theorem becomes false if the hypothesis that f is continuous is replaced with f being montone, or with f being strictly monotone. (can use the same counterexample for both)

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

The intermediate value theorem states that for a continuous function F on a closed interval [a,b] for any number k between F(a) and F(b), that is F(a)<k<F(b), there is at least on number c such that f(c) = k.

If we replace the word continuous with monotonic then the theorem does not hold.

Example: Keep in mind that monotonic means always increasing or always decreasing. I will use the function F(x) = 1/x because this function is always increasing over its entire domain. It as also not continuous at zero so let the interval be [-4,4] and k=0. There is no real number c such that F(c) = 1/c = 0. I am attaching a graph to show this.

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