Use the intermediate value theorem to prove that if $f:[0,1]\to [0,1]$ is continuous, then there exists $c\in [0,1]$ such that $f(c)=\sqrt{c}$

Suppose that $f(0)<f(1)$. Consider now, $f(0)<\sqrt{c}<f(1)$. By the intermediate value theorem, $\mathrm{\exists}b\in [0,1]:f(b)=\sqrt{c}$.

Now we need to show that its possible that $b=c$, how?

Suppose that $f(0)<f(1)$. Consider now, $f(0)<\sqrt{c}<f(1)$. By the intermediate value theorem, $\mathrm{\exists}b\in [0,1]:f(b)=\sqrt{c}$.

Now we need to show that its possible that $b=c$, how?