How can I show that function of $W^{1,p}(I)$ are absolutely continuous.

Question

Let $I\subset \mathbb R$ an interval of $\mathbb R$. How can I show that functions of $W^{1,p}(I)$ are absolutely continuous ? If this result is wrong, is there a result similar ?

Answer 1

A function $F$ is absolute continuous on $[a,b]$ if and only if there is a function $f$ s.t. $$F(x)-F(a)=\int_a^x f(t)\mathrm d t.$$

Let $u\in W^{1,p}((a,b))$. One can prove that there is a continuous function $\bar u\in \mathcal C(\bar I)$ s.t. $u=\bar u$ a.e. and $$\bar u(x)-\bar u(y)=\int_y^x u'(t)\mathrm d t,$$ for all $x,y\in I$. Therefore, $u$ is indeed absolutely continuous. Conversely, if $u$ is absolutely continuous and it's derivate is integrable over $(a,b)$, then it will be in $W^{1,p}((a,b))$.