Problem $1 .$ Show that the function $u: B(0,1) \rightarrow \mathbb{R},$ defined by $$u(x)=u\left(x_{1}, \ldots, x_{N}\right):=\left\{\begin{array}{ll} 1 & \text { if } x_{N}>0 \\ 0 & \text { if } x_{N}<0 \end{array}\right.$$ not belong to $W^{1, p}(B(0,1))$ for any $1 \leq p \leq \infty.$
Problem $2 .$ Let $\Omega \subset \mathbb{R}^{N}$ be an open bounded set, let $x_{0} \in \Omega,$ and let $1 \leq p<\infty .$ Prove that if $u \in C(\bar{\Omega}) \cap C^{1}\left(\Omega \backslash\left\{x_{0}\right\}\right)$ is such that the (classical) gradient $\nabla u$ belongs to $L^{p}\left(\Omega ; \mathbb{R}^{N}\right),$ then $u \in W^{1, p}(\Omega)$
For the first question,I have proved that with the dimension $1$, But I cannot extend my proof to the general case.
For the second problem,I have no idea about it.
Can someone help me solve these problems?Thanks in advance.