In the multivariate case, does an integrable characteristic function implies a continuous density function?

Question

In the univariate case, it is easy to see that if the characteristic function $\phi$ of a $\mathbb R$-random variable $X$ is integrable, then by the inversion theorem the p.d.f $f$ of $X$ is continuous. Does this result still hold for $X$ in $\mathbb R^d$, $d\geq2$ ?