if $\bar{X}\overset{p}{\rightarrow} \mu$ then $f(\bar{X})\overset{p}{\rightarrow} f(\mu)$?

Question

Is it true that if $\bar{X}\overset{p}{\rightarrow} \mu$ then $f(\bar{X})\overset{p}{\rightarrow} f(\mu)$ taking $f$ a continuous function and "$\overset{p}{\rightarrow}$" meaning "Convergence in Probability"? And if true, what would be a simple proof.

And the general case is forward too? This is:

if $X\overset{p}{\rightarrow} Y$ then $f(X)\overset{p}{\rightarrow} f(Y)$ taking $f$ a continuous function?