Suppose $f:[a,b]\to\mathbb{R}$ has a right limit $f(x+)$ at all $a \le x \lt b$ and a left limit $f(x-)$ at all $a\lt x\le b$.
Is the function $$g[a,b]\to\mathbb{R}:x\mapsto\begin{cases}f(x+)&a\le x\lt b\\ f(b)&x=b\end{cases}$$ càdlàg? and is the set $\lbrace x\in[a,b]:f(x)\neq g(x)\rbrace$ at most countable (like you would expect)?