In the Sion's minimax thorem we assume that $\ X$ is a compact convex subset of a linear topological space and $\ Y$ is a convex subset of a linear topological space. In addition we say that $\ f$ is a real-valued function s.t.
$\ f(x,\cdot)$ upper semicontinuous and quasiconvex function on $\ Y$, $\forall x\in X$, and
$\ f(\cdot,y)$ lower semicontinuous and quasi-convex on $\ X$, $\forall y\in Y$
Then $\min_{x\in X}\sup_{y\in Y} f(x,y)=\sup_{y\in Y}\min_{x\in X}f(x,y).$
I can understand why compcatness is needed (we can choose X and Y to be the real plane and take any increase function as f)
But I don't see why convexity and semi-continuity are neccessary?