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!
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A function in $W^{1,1}(I;X)$ is continuous on an interval $I$ with values in the Banach space $X$, see Lemma 7.1 in the book "Nonlinear partial differential equations" by Roubicek. Just choose $I=[a,b]$ and $X=\mathbb{R}^n$, and remember $W^{1,\infty}(I;X) \subset W^{1,1}(I;X)$. – Cahn Dec 17 '19 at 20:07
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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.
Jonas Lenz
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