Is $f$ differentiable on $[a,b]$ and $f(a)=a$ and $f(b)=b$ then $f'$ integrable on $[a,b]$?
I have just found that there is a differential function $f$ such that $f'$ isn't continuous at $x=0$. I wonder if $\int_a^bf'(t)dt=f(b)-f(a)$.
Is $f$ differentiable on $[a,b]$ and $f(a)=a$ and $f(b)=b$ then $f'$ is integrable on $[a,b]$?
0
$\begingroup$
calculus
definite-integrals
-
0Hint: What condition(s) on $f'$ need to hold if $f'$ is to be integrable? Are these conditions guaranteed by differentiability of $f$? – 2017-01-20
1 Answers
1
The derivative of an everywhere differentiable function $f$ is not necessarily Riemann integrable, even if $f'$ is bounded. In such cases the set of discontinuity of $f'$ has positive Lebesgue measure.
See here