Infinitely many discontinuous points Give an example of a function $f: [0,1] \to \mathbb R$ with infinitely many discontinuous points. You can use the fact that:
$\displaystyle\lim\limits_{k\to\infty}$ $\displaystyle\sum_{i=1}^k \displaystyle\frac {1}{2^i} = 1$
I don't even know how to start. 

I don't know what the hint is trying to tell you, but here is a function that is discontinuous everywhere:
$$
f(x) = \cases{0 & if $x$ is rational\\1 & if $x$ is irrational}
$$
