when someone says $f(x,y)$ is increasing in $x$, do they mean partial differentiation $\frac{\partial f(x,y)}{\partial x}$ or total differentiation $\frac{d f(x,y)}{d x}$?
If $y=h(x)$, then $f(x,y) = f(x,h(x))$ increases in $x$ in a partial sense, but $f(x,y)$ may decrease in a total sense.
"$f(x,y)$ is strictly increasing in $y$" means that, for every $y_1, y_2$, and $x$ we have
$$y_1<y_2\Rightarrow f(x,y_1)<f(x,y_2)$$