I suddenly remembered a fact about absolutely continous functions, but I can't remember the proof or the book where I read about this. Could you help me?
An absolutely continous function on $(a,b)$ can be extended to an absolutely continous function to $[a,b]$.
Clearly I just need to prove that absolutely continous functions on $(a,b)$ are bounded. How do I prove this?
By definition, you know that the sum $$ \sum |f(x_i)-f(x_{i+1})| $$ can be made arbitrarily small (say $<1$) by choosing the sum $$ \sum |x_i-x_{i+1}| $$ small enough, say $<\delta$. Pick a partition of $(a,b)$: $\{a<x_1<x_2<\dots<x_N<b\}$ (uniform, for example) such that any point $x$ of $(a,b)$ is within a distance $<\delta$ from some point in the partition.
Given $x\in (a,b)$, choose the closest point $x_N$. Then, since
$$
|x-x_N|<\delta
$$
you have
$$
|f(x)-f(x_N)|<1
$$
which implies that, for every $x$
$$
\min_N f(x_N)-1<f(x)<\max_N f(x_N)+1,
$$
thus proving boundedness.
