Is there a function on $[a,b]$ that has a bounded derivative on $(a,b)$, is NOT continuous at $a$ and $b$, and is NOT of bounded variation on $[a,b]$?

Question

It is well known that a continuous function on a compact interval $[a,b]$ that has a bounded derivative on $(a,b)$ is of bounded variation on $[a,b]$. I am curious that whether the continuity at the endpoints $a$ and $b$ is essential for the function to be of bounded variation on $[a,b]$? In other words, is there a function on $[a,b]$ that has a bounded derivative on $(a,b)$, is NOT continuous at the endpoints $a$ and $b$, and is NOT of bounded variation on $[a,b]$?