Let $X$ be a compact metric space, and let $C(X)$ be the Banach space of continuous real-valued function on $X$, with the maximum norm.

Suppose $S\subset C(X)$ is a set of functions with the following property:

For every ball $B(a,r)\subset X$ and for every $\epsilon>0$, there exists a function $f\in S$ such that $f(a)=1$ and $|f(x)|<\epsilon$ for $x$ outside the ball $B(a,r)$.

My question: does the above assumption on $S$ imply that the set $S$ spans $C(X)$, that is that every continuous function on $X$ can be arbitrarily approximated (in the max norm) by finite linear combinations of functions in $S$?

(An answer in the special case $X=[0,1]$ would also be of interest to me).