Give an example of a function which is continuous on $\mathbb{Q}\cap[2,4]$ but not uniformly continuous on the same set.

Question

Give an example of a function which is continuous on $\mathbb{Q}\cap[2,4]$ but not uniformly continuous on the same set.

I'm actually pretty lost with this one. I think I need to create a function that will generate irrationals or something? Any help will be appreciated.

As a side note, this should be solved really using only the $\epsilon$-$\delta$ definition of uniform continuity.

Answer 1

Go with the function $f(x)=\begin{cases}1&\text{if }x^2>5\\ 0&\text{if }x^2<5\end{cases}$. The function takes finitely many values, and the preimage of points is closed. However, it is apparent that $f$ is not uniformly continuous.