Consider the following two functions, one of which is continuous and the other is absolutely continuous. I have problem to understand the definition graphically. So I wondered if I can have a graphical elaboration.
$1) f(x)=x \sin(\frac{1}{x})$
$2) f(x) = \sqrt{x}$
On
It is not a concept that lends itself well to a graphical interpretation.
Let $o_n = {1 \over 2 \pi n + { \pi \over 2}}$, $z_n = {1 \over 2 \pi n}$. Note that $f(o_n) = {1 \over 2 \pi n}, f(z_n ) = 0$.
It is easy to see that $\sum_n (z_n-o_n) < \infty$, so for any $\delta>0$ we can find some $N$ such that $\sum_{n \ge N} (z_n-o_n) < \delta$.
However, for any $N$ we see that $\sum_{n \ge N} |f(z_n)-f(o_n)| = \sum_{n \ge N} {1 \over 2 \pi n} = \infty$.
Not a particularly graphical answer, but absolute continuity is similar to uniform continuity, if you have intuition for that. The difference is that absolute continuity requires the $\epsilon\mapsto\delta$ mapping to "work" even when you "split up" the $\delta$-sized interval into (finitely many) disjoint "mini-intervals" whose lengths sum to $<\delta$. The corresponding "mini-epsilons" still must sum to $<\epsilon$.