# Does $f(x)$ exist such that $f(x)$ can't be integrated but $f'(x)$ can?

I'm looking for an example of a function (if such a function exists) that cannot be integrated, but its derivative can.

Also, Does such a positive function exist, such that its co-domain is always positive? If so, do you have an example of one?

Thank you

• No. The existence of derivatives implies continuity. Continuity implies integrability.
– A.S
Apr 13, 2013 at 6:58
• Lebesgue integral always improves the regularity of an integrand. Thus such thing cannot happen. Apr 13, 2013 at 6:58

• I can see what you've said is true if the domain is a closed interval, but what if $f$ is defined on an open interval? Apr 13, 2013 at 7:26
• Even if it is closed you have to be pricese: Take the function $f : \mathbb{R} \to \mathbb{R} : x \mapsto 1$. It is not even Lebesgue integrable on $\mathbb{R}$. The theorem you state is true for bounded intervals. Then again Riemann integration only works for functions defined on a bounded domain. Apr 13, 2013 at 7:34
• For open domains this is not true. Take for example $f: ]0,1] \to \mathbb{R}: x \mapsto \dfrac{1}{x}$. Apr 13, 2013 at 7:35