Let $X$ be a reflexive and separable Banach space. Assume that $f\colon [0,b]\to X$ is absolutely continuous, so it has bounded variation
$$V(f)(b)=\sup \{V(f,P)\,|\, P\quad\text{is partition of}\quad [0,b]\},$$
where $V(f,P)=\sum_{k=1}^{m}\|f(x_k)-f(x_{k-1})\|_X$. One can easily deduce that the function $t\mapsto V(f)(t)$ is increasing for $t\in [0,b]$. But, is the function $V(f)(\cdot)$ differentiable???
Also, how to prove that for $0\le t \le t+h\le b $ we have $$\|f(t+h)-f(t)\|_X\le V(f)(t+h)-V(f)(t)\quad ???$$
Best Answer
I don't see difference between the case when $X=\mathbb R$ and the general case. Even in the special case $X=\mathbb R$, $t \to V(f)(t)$ need not be differentiable at every point, but it is differentiable almost everywhere. The fact that this function is differentiable almost everywhere in the general case follows from the fact that all monotone functions are almost everywhere. The last inequality is also proved exactly as in the case of real valued functions.