What is the difference between Newton integrable function and Riemann integrable function?

Question

I am supposed to determime the difference between Newton integrable function and Riemann integrable function.

I know that A function $f : (a, b) \to \Bbb R$ is said to be Newton integrable if $f$ has a primitive $F$ in $(a, b)$, and if the one sided limits $F(a_+)$ and $F(b_−)$ exist and are finite.

I also understand what is Riemann integral: I have to make partition on the graph and the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer.

But somehow I am unable to distinguish the difference between those two, althougtht there are the definition.

Is somehow the area below the graph different or what is the difference?

Thanks.

Answer 1

Take for example $f$ from $[-1,1]$ to $\Bbb R$ given by $$f(x)=-1 \text{ if } x<0$$ and $$f(x)=1 \text{ if } x\ge 0$$

then $f$ is Riemann integrable since it has only one discontiuity point,

But

It is not Newton integrable since it has no primitive at $(-1,1)$.