Is the function $f\colon{\mathbb{Z}}\to\overline{\mathbb{Z}}, x\mapsto x+\infty$ continuous?

Question

Is the function $f\colon \mathbb{Z}\to\overline{\mathbb{Z}}, x\mapsto x+\infty$ continuous?

Here, $\overline{\mathbb{Z}}=\mathbb{Z}\cup\{\pm\infty\}$.

I am not sure how to answer this question.

Answer 1

I'm going to interpret your function to be $f(x)=\infty$.

A constant function is continuous. Why? Consider an open set $O$ in $\bar{\mathbb{Z}}$:

• If $O$ contains $\infty$ then its preimage is ${\mathbb{Z}}$, which is an open set by the axioms of topology.
• If $O$ does not contain $\infty$, then its preimage is $\emptyset$, which is an open set by the axioms of topology.

By the definition of a continuous function in topology, $f$ is continuous. We don't need to know anything about the topology on $\bar{\mathbb Z}$.