Question

If f(x) is continuous and 0 <= f(x) <= 1 for all x on the interval [0,1], then is it true that for some number x, f(x) = x?

Answer 1

Yes, there must be a `c` in `[0,1]` with `f(c)=c`.

Explanation:

If `f(0)=0`, we're done, so suppose `f(0) > 0`.

If `f(1)=1` we're done, so suppose `f(1)<1`.

Let `g(x) = f(x)-x` and apply the Intermediate Value Theorem to `g` to conclude that there is a `c` with `g(c)=0` so `f(c)=c`.