does the countinous function over continuous function is continuous?

Question

If I have a function $g:[0,1]\rightarrow [0,1]$ which is continuous and $f:\Omega \rightarrow [0,1]$ which is right continuous. Is $g(f)$ continuous or just right continuous? Here $\Omega$ is the event space and $f$ is a random variable.

Answer 1

In general it will only be right-continuous as we can take $g(x)=x$ and then $g\circ f=f$.