More formally, let a surface be of the form $z=f(x,y)+g(x,y)$. Let $0\leq f(x,y)\leq h(x)$ and $0\leq g(x,y) \leq k(y)$ (running out of variables!) Is it valid to say that $0\leq f(x,y)+g(x,y)\leq h(x)+k(y)$?
2026-04-08 22:39:45.1775687985
Can you sum the bounds of two separate surfaces to bound the new summed surface?
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I don't see what you gain from thinking of your functions as surfaces. Think of them as real numbers: for any given $x$, $y$ obviously $0 \leq f \leq h, \ 0\leq g \leq k \Rightarrow f + g \leq h + k$
UPD: Can't comment :( About your example: yes, $h+k \rightarrow \infty$ at 0, but so do any bounds on $f$ and $g$ (say, $f \leq 1/x^2$), and the inequality still holds at any point.