Let $\mathbb{Z}$ be the integers with the discrete topology. Let $f: \mathbb{Z} \to \mathbb{R}$. Give conditions to have $f$ continuous.

Question

This is part of an exercise with different sections. I've proved that $\mathbb{Z}\subset \mathbb{R}$ has the discrete topology as the subspace topology. Now they ask me the following:

Let $\mathbb{Z}$ be the integers with the discrete topology. Let $f: \mathbb{Z} \to \mathbb{R}$. Give conditions to have $f$ continuous.

My aim: it sounds pretty ridiculous, but I think that every function is continuous. All open sets on $\mathbb{R}$ go to somewhere open through the inverse of $f$, even if they go to the empty set. Every possible combination of elements on $\mathbb{Z}$ is open. So every function defined there is continuous, no matter what.

Am I right or there is any mistake I'm missing?

Thanks for your time.

Answer 1

You are correct.

If $X$ has the discrete topology, any subset is open, therefore any map $f:X\to Y$ is continuous.

Answer 2

You can take $\delta = 1$ for any $\varepsilon >0$, so indeed even simple emtric criteria will give that all functions are continuous.