Is a a small perturbation of proper map still proper?

Question

Let $X,Y$ be two Banach spaces. Let $f:X\to Y $ be a $C^2$ map.

Suppose in addition that $f$ is also proper i.e. $f^{-1}(K)$ is compact for any $K\subset Y$ compact (this is equivalent to $f^{-1}(y)$ compact for any $y\in Y$ and $f$ closed).

Question: is it true that a sufficiently small perturbation of $f$ is still proper?

More precisely, is it true that for any $F\in C^k(X \times[0,1]\to Y)$, with $F_0 = f$, there exists $T>0$ such that $F_t$ is proper for any $t\leq T$? In my mind $k>1$, but I would be curious to know also about counterexamples in the $k\leq 1$ cases.

Answer 1

No. Counterexample: Let $X = Y = \mathbb R$, let $g: \mathbb R \to \mathbb R$ be a smooth non-decreasing function so that $$g(x) = \begin{cases} x & \text{ if } |x| < 1 \\ 2 & \text{ if } x \ge 2 \\ -2 & \text{ if } x\le -2.\end{cases}$$

Then define $F: [0,1] \times \mathbb R \to \mathbb R$ by $F(0, x) =x$ and $F(t, x) = t^{-1}g(tx)$ when $t\neq 0$.

Clearly $F$ is smooth in $(0,1) \times \mathbb R$. At each $(0,x)$, note that $F(t, x) = x$ whenever $t|x| <1$. Thus $F$ is also smooth at $(0,x)$.

Note that $F(0, \cdot)$ is proper but $F(t, \cdot)$ is not proper for all nonzero $t$.