Is a $W^{1,\infty}$ function a continuous function?

Question

Let consider a function $f\in W^{1,\infty}([a,b];\mathbb{R}^n)$.

Somebody can suggest me a reference where I could confirm if $f$ is a continuous function, due to $f \in W^{1,\infty}([a,b];\mathbb{R}^n)$?

Thank you very much!

Ana

Answer 1

You may have look for example at the book "Functional Analysis, Sobolev Spaces and PDEs" by Brezis, especially at Chapter 9.

Generally, one has that if $\Omega \subset \mathbb{R}^n$ is a bounded domain and "smooth enough", then $f \in W^{1,p}(\Omega;\mathbb{R}^n)$ with $p>n$ has a continuous representative.