Let $E$ be a normed vector space(or Banach space), $\Omega$ an open set of $\mathbb{R}\times E$ and $f$ a function from $\Omega$ to $E$. We say that $f$ is Lipschitz in the second variable on a subset $W$ of $\Omega$ if there is a constant $k$ such that :
$$(t, x_1), (t, x_2) \in W \Rightarrow ||f(t, x_1) - f(t, x_2)||\leq k||x_1 - x_2||$$
We say that $f$ is locally Lipschitz in the second variable if all point of $\Omega$ has a neighborhood on which $f$ is Lipschitz in the second variable.
Now we assume that $f$ is locally Lipschitz in the second variable, $K$ is a compact set include in $\Omega$. Then $f$ is Lipschitz in the second variable on a neighborhood of $K$.
Can someone help？ We may assume that $\Omega$ is connected if it's needed.
HINT: Use the fact that $K$ can be cover by a finite number of neighborhood on which $f$ is Lipschitz in the second variable.