While it would seem to be true, I have found that there are examples of functions that are differentiable but not Riemann Integrable. Would a step function, for example, be integrable but not Riemann Integrable? So would all differentiable functions still be integrable in the general sense?
[Math] Does every differentiable function have integrable derivative
derivativesintegration
Best Answer
If you take integrability in the sense of Henstock-Kurzweil then the answer to the question of the title of your post is yes! The amazing thing about the Henstock-Kuzweil integral is then that the fundamental theorem of calculus is valid for every differentiable function, without further assumptions: let $$ f\colon [a,b]\rightarrow\mathbf R $$ be a differentiable function. Then $f'$ is HK-integrable and $$ \int_a^bf'=f(b)-f(a). $$ The proof is very easy and a true marvel, showing that the HK-integral deserves much more attention than it gets.