Infimum of a function of two variables.

Question

The following result holds true:

For a real function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ we have that $\inf_{x,y} F(x,y)\leq \inf_x (\inf_y F(x,y)) $.

My question is on whether or not it is possible to provide an example where we have strict inequality? If so, I would be interested in seeing an example. If not, why?

Thanks in advance.

Answer 1

In fact, given any two non-empty sets $X,\,Y$ and $f:X\times Y\to\mathbb{R},$ we have equality since: $$ \forall (x,y)\in X\times Y:\,f(x,y)\geq\inf_{y}f(x,y)\geq\inf_{x}\left(\inf_{y}f(x,y)\right) \\ \implies \inf_{x,y} f(x,y)\geq\inf_{x}\left(\inf_{y}f(x,y)\right) $$ While inequality in the other direction holds since: $$ \forall x\in X:\,\inf_{y}f(x,y)\geq \inf_{x,y}f(x,y)\implies \inf_{x}\left(\inf_{y}f(x,y)\right)\geq \inf_{x,y}f(x,y). $$ This argument can be easily generalised to take the infimum of a real-valued function with $N$ arguments.