1. ## Continuity Problem

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.

2. Originally Posted by Sterwine
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 $c\in\mathbb{Q}$, then $\forall\delta>0$, $\exists x$ s.t. $|c-x|<\delta$ and $|f(c)-f(x)|=c^2$. So given a point $c$, letting $\epsilon=\frac{c^2}{2}>0$ (for instance), will ensure that $|c-x|<\delta$ and $|f(c)-f(x)|\geq\epsilon$.

You can do a very similar thing for $c$ irrational.