I'm having difficulty with proving directly that every metric space is a Tychonoff space. I can show that every metric space is Hausdorff.

Let $\displaystyle (X,d)$ be a metric space, let $\displaystyle A$ be a closed subset of $\displaystyle X$, let $\displaystyle y \in X-A$. Hint: Define $\displaystyle f: X \rightarrow I$ by $\displaystyle f(x)= min\{d(x,A)/d(y,A),1\}$. So, $\displaystyle f(y)=1$ and $\displaystyle f(x)=0$ for every $\displaystyle x \in A$. I am stuck on proving that this function is continuous. Anyone can help me?