I am searching for a monotonically increasing and invertible function in $2$ variables. I know several monotonically increasing functions. This is also true for invertible functions. But I am searching for a function which is both monotonically increasing and invertible in $2$ variables.
Eg. a function of the form: $z=f(x,y)$ , here x<=y such that we can get $x=f^{-1}(z)$ and $y=f^{-1}(z)$ Also, if $x_1 \lt x_2$ then $z_1 \lt z_2$. In case $x_1=x_2$ then if $y_1\lt y_2$ then $z_1\lt z_2$.
Please help me figure out some function of this type.