Suppose I have a quasiconvex function $g:\mathbb{R} \rightarrow \mathbb{R}$ and a step-function $f(x)$ defined as: \begin{align} f(x) = \begin{cases} a \mbox{ if } x\geq \alpha \\ b \mbox{ if } x< \alpha \end{cases}, \end{align} for some $a,b,\alpha \in \mathbb{R}$.
Suppose I take the pointwise minimum $h(x)=\min\{g(x),f(x)\}$. Now, I know that in general the pointwise minimum with a quasiconvex function is not quasiconvex, but I believe that under the additional assumption that $h(x)$ is continuous, that $h(x)$ should be quasiconvex as well. This continuity assumption prevents all kinds of trivial counterexamples for $h(x)$ being quasiconvex that I could think of.
Who has an idea how to prove this, or has a counterexample for me?
Thank you very much!