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!!

1 Answer | Add Yours

Top Answer

embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

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.

-----------------------------------------------------------------

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

----------------------------------------------------------------

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].

We’ve answered 318,911 questions. We can answer yours, too.

Ask a question