Inequality with $W^{1,p}$ norm

Question

For $a<b \in \mathbb{R}$, let $G = (a,b)$. How can I show that firstly for every $v \in W^{1,p}(G)$ there exists a unique $\tilde{v} \in C^0(\overline{G})$ such that for almost every $x \in G$ it holds that $v(x) = \tilde{v}(x)$? And secondly that for every $p \in [1,\infty]$ there exists a constant $C>0$ such that $$ \lVert \tilde{v} \rVert_{C^0(\overline{G})} \leq C \lVert v \rVert_{W^{1,p}(G)}$$ for every $v \in W^{1,p}(G)$?

Answer 1

Both claims hold since $W^{1,p}(G)\hookrightarrow C^0(\overline{G}).$ Just look for Sobolev embeddings and note that $\text{dim}G=1$.