An exercise in Royden & Fitzpatrick asks to show that if g is integrable on [a,b] and we define the antiderivative of g as: $f(x)=\int_{a}^{x} g(x)$ for all $x\in[a,b]$, then f is differentiable almost everywhere on (a,b).
Of course, we want to use the Lebesgue Theorem: $f$ monotone on $(a,b)=>f$ diff a.e. on $(a,b)$. But I'm having trouble with the logic of showing monotonicity. I want for any $x>y$ in $(a,b)$, $f(x)>f(y)$ or $f(x)<f(y)$,for either increasing or decreasing. I have:
$\int_{a}^{x} g(x)-\int_{a}^{y} g(x)=\int_{y}^{x}g(x)$. Now, if g(x)=0, then $\int_{y}^{x}g(x)=0$. Otherwise we have inequality in either direction and we still get monotonicity. Is this correct?
Thanks for the help.