Edited version of question:
Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ is cadlag (that is, it is right continuous and its left limits exist) and $g:\mathbb{R}\longrightarrow \mathbb{R}$ is continuous. Then is it necessarily true that $g\circ f:\mathbb{R} \longrightarrow \mathbb{R}$ is also cadlag?
I am trying to prove this more general statement for fun, but I have been quite stuck for a while now.
PS: Currently for what I want to prove all I really need is the weaker statement that $g$ to be the mapping $x\mapsto x^2$. However I am curious whether this more general statement holds as I am sure it will become handy later on/it is in general a useful fact to know.
Many thanks to spaceisdarkgreen for pointing out my horrendous mistake!