Draft:Karamardian's anomaly

In convex analysis, Karamardian's anomaly is an example of a pathological function that is strictly ray-quasiconvex but not quasiconvex. It is defined as follows for all $x\in \mathbb {R}$:

$$f(x)=\left\{{\begin{array}{lll}1&{\text{if}}&x=0\\0&{\text{if}}&x\neq 0\end{array}}\right.$$

To show that it is strictly ray-quasiconvex, the property $f(\lambda x+(1-\lambda )y)<\max\{f(x),f(y)\}$ must hold when $f(x)\neq f(y)$ and $0<\lambda <1$. Since $f(x)\neq f(y)$, at least one of $x$ and $y$ is necessarily $0$, and hence $\max\{f(x),f(y)\}=1$.

To show that it is not quasiconvex, take $x=-1$, $y=1$ and $\lambda ={\frac {1}{2}}$. Then $f(\lambda x+(1-\lambda )y)=f(0)=1>0=\max\{f(x),f(y)\}$, contradicting that $f(\lambda x+(1-\lambda )y)\leq \max\{f(x),f(y)\}$.