I think a function having removable discontinuities is continuous at almost every point of an interval $[a, b]$.
For example, let $f$ be a function by \begin{align}f(x)=\begin{cases}\sin(x) & x\in\mathbb{R}\setminus\{0\} \\ 10 & x=0\end{cases}\end{align}
How about a function having jump discontinuities?
For example, let $g$ be a function by \begin{align}g(x)=\begin{cases}x^2 & x\lt0 \\ 1 & x=0 \\2-x^2 &x\gt 0\end{cases}\end{align}
Is this function $g$ also continuous at almost every point of an interval $[-1, 1]$?