Define h: R-> R by
h(x) = { x^2, x is rational
{ 0, x is irrational
Prove that h is not continous at c not equal to 0. Split the proof into two cases: one for c rational, the other for c irrational.
Well if $\displaystyle c\in\mathbb{Q}$, then $\displaystyle \forall\delta>0$, $\displaystyle \exists x$ s.t. $\displaystyle |c-x|<\delta$ and $\displaystyle |f(c)-f(x)|=c^2$. So given a point $\displaystyle c$, letting $\displaystyle \epsilon=\frac{c^2}{2}>0$ (for instance), will ensure that $\displaystyle |c-x|<\delta$ and $\displaystyle |f(c)-f(x)|\geq\epsilon$.
You can do a very similar thing for $\displaystyle c$ irrational.