Piecewise $C^1$ function is element of $W^{1,\infty}$

Question

Hey I'm confused about the following (apparantly) fact:

Let $u:[a,b]\to\mathbb{R}$ a piecewise $C^1$ function, i.e. there exists

$a=t_1<t_2<\cdots < t_n = b$ such that $u|_{[t_i,t_{i+1}]} \in C^1([t_i,t_{i+1}])$

Then $u\in W^{1,\infty}(a,b)$. So then I have to show that $$\sup_{(a,b)}|u|+ \sup_{(a,b)}|u'|$$ exist.

$\sup_{(a,b)}|u|<\infty$ is clear ofcourse, but I dont understand how $\sup_{(a,b)}|u'|<\infty$ can hold if at the points $t_i$ the $u'$ is not defined. We can only talk about left- and right derivatives. So how does this work then? Or am I misinterpreting the question?

Answer 1

For $u \in W^{1,\infty}$, you need to show that $u'$ is actually the weak derivative. The weak derivative itself is a function from $L^\infty$, i.e., it needs to be defined only a.e.

Hence, the non-existence of the strong derivative at $t_i$ does not matter.