Extension of a compact set via continuous functions

Question

Define a set $\mathcal{D}\subset \mathbb{R}^n$ and a continuously differentiable function $h:\mathcal{D} \to \mathbb{R}$. Consider the set $\mathcal{C} = \{ x \in \mathcal{D} : h(x) \geq 0\}$. Let $h$ be defined on $\mathcal{D}$ such that $\mathcal{C} \subset \mathcal{D}$, and is compact.

Now consider the following set: $$ \mathcal{C}_e = \{ x \in \mathcal{D} : h(x) \geq - \epsilon \} \subset \mathcal{D}, $$ for some $\epsilon >0$.

What can we say about $\mathcal{C}_e$. Is it compact as well?

Answer 1

It is not true. Here is a counter example.

Counter example for the compactness arguement

Answer 2

$C_e$ is closed since it is the primage of $h^{-1}[\epsilon,\infty)$ and a closed subset of a compact set is again compact (in this case, since it is bounded.)