consistency of inverse cdf

Question

If $\hat{F}(x)$ is a consistent estimator of $F(x)$ where $F(x)$ is a cdf, can we state that $\hat{F}^{-1}(x)$ is also a consistent estimator of $F^{-1}(x)$? Is that straightforward? Why? You can make any assumption you want if that is necessary. I am looking for a discussion.