Proving that for a continuous random variable the probability at a single point is zero

Question

I wish to prove that for any continuous random variable $X$, for any point $a$ we have $P(X=a)=0$.

I have got this far:

Let $f_x(x)$ denote the density function of $f$.

$$\lim_{\epsilon \rightarrow 0}P(a-\epsilon<X<a+\epsilon)=\int_{a-\epsilon}^{a+\epsilon}f_x(u)du=0$$ by continuity of $f_x$ and the fact that an integral with equal limits of integration is $0$. However, apparently, this operates on the assumption that $f$ is bounded. How do I get around this?