# True of false: If f(x) is continuous and 0<=f(x)<=1 for all x in the interval [0,1], then for some number x, f(x)=x. Explain your answer.Thanks!!

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We are given that `f(x)` is continuous, and `0<=f(x)<=1` for `x in [0,1]`

(1) Suppose `f(x)<x` for all x in [0,1]. This contradicts `0<=f(x)` at x=0.

(2) Suppose `f(x)>x` for all x in [0,1]. This contradicts `f(x)<=1` at x=1.

Since f(x) is not less than every x nor greater than every x in the interval, it must equal some x in the interval.

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**There exists an `x in [0,1]` such that `f(x)=x` **

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Intuitively, draw the line y=x on the interval [0,1]. The graph of f(x) cannot be above the line on the whole interval; what happens at x=1 then? The graph of f(x) cannot be below the line on the whole interval; what happens at x=0? So teh graph crosses the line somewhere.

This is only true because f(x) is continuous -- if f(x) is not continuous you could have f(x)=1 for [0,1/2) and f(x)=0 for [1/2,1].