The following true or false question is taken by an Integral Calculus Exam but I cannot come up with a counter example.
Question: If a function has only one point of discontinuity , then it is Riemann integrable and its indefinite integral is a differentiable function.
Now, since our function has only point of discontinuity this means that it is Riemann integral. Let us call $F$ the antiderivative and let $x_0$ be the point of discontinuity of our function. Clearly $F$ cannot be differentiable at $x_0$ since $f$ is not continuous there. Do we know anything else about $F$? I think the statement is false but I cannot come up with a counter example. Any help?