Domain Differentiation question

[Bachelier]

if a function f is differentiable of [0,2pi] can I integrate its derivative df on [-pi, pi]?

 

[Vorde]

First of all, df is not the derivative of f, it is the differential of f, $\frac{df}{dx}$ is the derivative of x (assuming f is a function of x).

Second of all, why would you think that it's derivative would have a different domain?

 

[HallsofIvy]

The function, f(x)= 0 if x< 0 and rational or $x> 2\pi$ and rational, f(x)= 1 if x< 0 and irrational or $x> 2\pi$ and irrational, f(x)= 3 for all other x, is differentiable for all x in $[0, 2\pi]$ but not differentiable for any other f. Since it does not have a derivative on $[-\pi, 0]$, no, its derivative cannot be integegrated there!