Suppose we have functions $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$, and $f$ is left continuous and $g$ is right continuous. I was wondering if it is always the case that:
\begin{align*} \lim_{y\to y_{0}^{-}}(f\circ g)(y) = f(\lim_{y\to y_{0}^{-}} g(y) ) \end{align*}
and
\begin{align*} \lim_{x\to x_{0}^{+}}(g\circ f)(x) = g(\lim_{x\to x_{0}^{+}} f(x_{0}) ) \end{align*}
It may not be the case, because the function inside can "reverse the order". For example let $g(x) = -x$ and let $f(x) = 0$ for $x\in (-\infty,0]$ and $f(x)=1$ for $x>0$. Then $$\lim\limits_{x\to 0^-} f(g(x)) = \lim\limits_{x\to 0^-} f(-x) = 1 \neq f(0).$$
Also note that if $g$ is only right-continuous, the limit $$\lim\limits_{y\to y_0^-} g(y)$$ need not exist.