When $F$ is not everywhere differentiable but almost everywhere differentiable, can we still write:
$\int_a^bF'(x)dx=F(b)-F(a)?$
In my opinion there is no problem since $F'(x)$ does not exist for only a measure zero set, which cannot affect the result of integration. To formalize this, do we need to use Lebesgue integration?
Best Answer
Consider $$ F(x)=\left\{ \begin{array}{ccc} 0 & \text{if} & x\in [-1,0], \\ 1 & \text{if} & x\in (0,1], \\ \end{array} \right. $$ Then $F$ is differentiable everywhere but $x=0$, and $F'(x)=0$, $F'$ is a measurable function, which is Lebesgue integrable and $$ \int_{-1}^1 F'(x)\,dx=0\ne F(1)-F(-1)=1. $$
However, if $F$ is absolutely continuous, then $F$ is differentiable almost everywhere and $\int_a^b F'\,dx=F(b)-F(a)$.