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

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.

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,
It is not Newton integrable since it has no primitive at $$(-1,1)$$.
• It does have a primitive, $|x|$ Commented Sep 21, 2019 at 22:32