# 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?

Yes, there must be a $c$ in $\left[0 , 1\right]$ with $f \left(c\right) = c$.
If $f \left(0\right) = 0$, we're done, so suppose $f \left(0\right) > 0$.
If $f \left(1\right) = 1$ we're done, so suppose $f \left(1\right) < 1$.
Let $g \left(x\right) = f \left(x\right) - x$ and apply the Intermediate Value Theorem to $g$ to conclude that there is a $c$ with $g \left(c\right) = 0$ so $f \left(c\right) = c$.