Real valued continuous function on arbitrary space

Question

I was wondering if we can construct a continuous function $f:X \rightarrow \mathbb{R}$ that is not the constant function. Here X is an arbitrary infinite set with some metric d defined on it. The metric on $\mathbb{R}$ is the euclidean metric. I am solving a problem where I need to use the fact that such a function exists but I have no clue where to begin.

Edit: I have modified the problem to include a metric d on X.

Your question is very large. What is $X$? Talking about continuity requires a topology on $X$. If no topology is mentioned, there can be no answer — Didier, May 28 '20 at 13:07
I was thinking the same thing. So a topology is necessarily required? — Suraj, May 28 '20 at 13:12
If $X$ is an arbitrary set and You equip it with the trivial topology, the constant functions are the only continuous ones — Peter Melech, May 28 '20 at 13:17

score 2 · Answer 1 · answered May 28 '20 at 13:35

2

Fix some $a \in X$. Then $f(x) = d(a,x)$ is a real-valued continuous function on $X$, which is non-constant if $X$ has more than one member.

answered May 28 '20 at 13:35

Robert Israel

448,999

Ahh. I should have been able to do this. So simple! – Suraj May 28 '20 at 13:39
@RobertIsrael Hi professor Israel, could I ask your assistance here? – Antonio Maria Di Mauro May 28 '20 at 16:24

Real valued continuous function on arbitrary space

1 Answers1