Example of a non-quasiconvex function which is quasiconvex at a point

Question

I have found examples of non-convex functions which are convex at some points.

The function $$f(x) \mapsto \begin{cases} x^2 & x < 1 \\ 1 & x \geq 1 \end{cases} $$ is non-convex, but is convex for $x\leq-1$.

So far I have not been able to find non-quasiconvex funtions which are quasiconvex at some/a point(s).

By quasiconvexity at a point I understand this:

A function $f: \mathcal{X} \to \mathbb{R}$ is quasiconvex at $\mathbf{x'} \in \mathcal{X}$ if $$f(\lambda \mathbf{x'} + (1- \lambda)\mathbf{x}) \leq \max\{f(\mathbf{x'}) , f(\mathbf{x})\}$$ for all $\mathbf{x} \in \mathcal{X}.$

Answer 1

Just found an example I think.

The function $$f(x) \mapsto \min \{1-\sqrt{\lvert x \rvert}, 0\}$$ is nonquasiconvex but is quasiconvex at points $\lvert x \rvert \leq 1$.