Does the set of Continuous functions always separate points and vanish nowhere?

Question

Let $X$ be an arbitrary metric space. Does $C(X;\mathbb{R})$ always separate points and vanish nowhere?

I think this is true because the functions $f(x) = x$ and $g(x) = 1$ are both in $C(X;\mathbb{R})$ and separate points and does not vanish.

Thank you for the help.

Answer 1

Yes, these two functions do suffice to show that for $X\subseteq \mathbb{R}$. Or use $f$ and $g(x)=x+1$, as $f$ and $g$ never simultaneously are $0$. But the definition of $f$ assumes all points of $X$ are reals.

For general metric $X$ consider using $f_p(x) = d(x,p)$ for $p \in X$ as continuous functions, and also $g$.