Support of a measure having continuous density with respect to Lebesgue measure

Question

Let $\Omega=R^d$ and let $\nu$ have a continuous density $f$ with respect to the Lebesgue measure. Show that the support of $\nu$ is the closure of ${\{f > 0}\}$

Definition of support used:

Any measurable set A with $\nu(A^c)=0$ is called a support of $\nu$.