Let $f: [a,b]\to R$ be a continuous function differentiable on $(a,b)$.
Show that for any $ε>0$, there are uncountably many points $x\in[a,b]$ such that $f′(x)\in(M−ε,M+ε)$
I know that by MVT, there exists $x^*\in(a,b)$ such that $f'(x^*)=M$. But I am stuck here. How should I proceed?
PS: this is a homework problem