Is $f:\mathbb{R}^{2} \backslash E \times\{0\}\to\mathbb{R}$ uniform continious function?

Question

Can you help me with this problem?

Let $E\subseteq \mathbb{R}$, $U=\mathbb{R}^{2} \backslash E \times\{0\}$. Prove that the following statements are equivalent:

1)Any continious function $f:U\to\mathbb{R}$ having limited partial derivatives outside $\mathbb{R}\times\{0\}$ is uniform continious function in $U$.

2)$E$ is totally disconnected set ($E$ hasn't open intervals)

I think if $E$ has intervals we should find not being a uniformly continuous function, but I can't do it.