Hausdorff spaces from continuous functions

Hausdorff spaces from continuous functions

The question is to prove a topological space is Hausdorff if for every $p$
in the space there exists a continuous function $f_{p}$ such that
$f^{-1}(0) = \{p\}.$ (The inverse here is implied as converse, not a
bijective inverse).
My thinking is that if we take the open subset of $\mathbb{R}$ $(-1;1)$
for each $f$ since the map is continuous we get open sets containing
$p_i,$ and it boils down to showing the intersection is empty. I can't
quite follow this important part through.
Thanks in advance.