Domain Differentiation question

if a function f is differentiable of [0,2pi] can I integrate its derivative df on [-pi, pi]?
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 Recognitions: Gold Member 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?
 Recognitions: Gold Member Science Advisor Staff Emeritus 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!

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