Can an integrable function on an interval have jump discontinuities?

Question

So I was wondering about this question

An integrable function on an interval cannot have jumps.

I always thought that a differentiable function on an interval cannot have jumps, so i was thinking that also an integrable function on an interval cannot have jumps.

Am i correct?

Moreover, If the derivative does not exist at a point, then this critical point cannot be either a local maximum or a local minimum. This question is false; correct me if im wrong

Answer 1

No. The function$$\begin{array}{ccc}[-1,1]&\longrightarrow&\mathbb R\\t&\mapsto&\begin{cases}0&\text{ if }t<0\\1&\text{ otherwise}\end{cases}\end{array}$$is Riemann-integrable.

Answer 2

It is well-known that a bounded function on a closed and bounded interval is Riemann integrable over that interval iff the set of points where it is discontinuous has measure $0$. A jump discontinuity happens at a single point, and a set containing a single point has measure $0$. The type of discontinuity is irrelevant to the theorem.