Problem: suppose function $f:[a,b] \to \mathbb{R}$ is a continuously differentiable function except on some $D = \{x_{\alpha}\}_{\alpha \in A} \subset (a,b)$, where $D$ is of measure zero. Can I still conclude by the fundamental theorem of calculus that
$$f(b) – f(a) = \int_a^b f'(x)dx = \sum_{\alpha \in A} f(x_{\alpha}) – f(x_{\alpha+1})$$
and does the second equality make sense?
I think I am confused on the definition of the Riemann integral. Technically speaking, if a function is not define on a set of measure zero, do we just ignore these points and compute the integral of the function on each of the intervals where it is defined? I mean the integral ignores values a functions takes on measure zero sets, but I just want to feel clearer about this. Any help is appreciated!
Best Answer
The well known Cantor function is (constant on each of the intervals of $[0,1]\setminus C$ and hence) continuously differentiable on $[0,1]\setminus C$ (where $C$ is the Cantor set) and its derivative is $0$ almost everywhere. This leaves us with no hope of recovering $f$ by integrating the derivative.