how to determine the infimum of a multivariate function?

Question

I have function $h(x,y)=f(x,y)+g(y)$.

What is the role of $g(y)$ in $inf_x(h(x,y))$?

Is $g(y)$ going to be zero or it will be in $inf_x(h(x,y))$ with no change?

Answer 1

If you fix a value of $y$, then $g(y)$ may be seen as a constant with respect to $x$ (not necessarily zero) and $$\inf_{x} h(x,y) = g(y)+\inf_{x} f(x,y).$$ (simply think of $h(x,y)$ as a one-variable function $h(.,y) : x \mapsto h(x,y)$ depending on a fixed parameter $y$).