- #1

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## Main Question or Discussion Point

This is still not clear to me. Here's the conjecture:

True or not? I don't know a proof, and I don't know a counter example.

Assume that [itex]f:[a,b]\to\mathbb{R}[/itex] is such function that it is differentiable at all points of its domain, and that

[tex]

\int\limits_{[a,b]}|f'(x)|dm(x) < \infty

[/tex]

holds, where the integral is the ordinary Lebesgue integral. Then also

[tex]

\int\limits_{[a,b]}f'(x)dm(x) = f(b)-f(a)

[/tex]

holds.

Assume that [itex]f:[a,b]\to\mathbb{R}[/itex] is such function that it is differentiable at all points of its domain, and that

[tex]

\int\limits_{[a,b]}|f'(x)|dm(x) < \infty

[/tex]

holds, where the integral is the ordinary Lebesgue integral. Then also

[tex]

\int\limits_{[a,b]}f'(x)dm(x) = f(b)-f(a)

[/tex]

holds.

True or not? I don't know a proof, and I don't know a counter example.