Compact set and continue function

Question

Let $f : \mathbb R \times \mathbb R^2 \to \mathbb R$ be a continuous function and $I \subset \mathbb R$ a closed interval.

$$E = \{(x, (y_1, y_2)) \in I \times \mathbb R^2 : f(x, (y_1, y_2)) = C\}$$ for a certain constant $C \in \mathbb R$.

Is $E$ compact or what are the other conditions for $E$ to be a compact set?

thanks

Answer 1

You can write $E$ as

$$E=f^{-1}[\{C\}]\cap(I\times\Bbb R^2)\;,$$

which makes it clear that $E$ is a closed subset of $\Bbb R^3$; thus, $E$ is compact if and only if it is bounded in $\Bbb R^3$.

Answer 2

In general, no, it is not compact. If, say, $f$ is the constant function $C$, then $E=I\times\Bbb R^2$.

However, if $f$ is a proper map, then, yes, $E$ will be compact.