How to prove or disprove this lemma? Lemma: For any real-valued continuous function $ f(x,y) $, where $ x \in R^m $ and $ y \in R^n $.
There are smooth scalar functions $ A(x)\geq 0 $, $ B(y)\geq 0 $, $ C(x)\geq 1 $ and $ D(x)\geq 1 $ such that
$ \left| f(x,y) \right| \leq A(x)+B(y) $
$ \left| f(x,y) \right| \leq C(x)D(y) $. 

Let $M(N) = \max \{ |f(x,y)|: \max(\|x\|,\|y\|) \le N+1 \}$ (which is finite because $\{(x,y): \max(\|x\|,\|y\|) \le N\}$ is compact).  There are smooth increasing functions $a(x)$ and $a(y)$ such that $a(n) > M(n)$ and 
$b(n) > M(n)$ for each nonnegative integer $n$.  Take $A(x) = a(\|x\|)$ and $B(y) = b(\|y\|)$ (that may not be smooth at $0$, so adjust it slightly).
Then for any $(x,y)$, if $\|x\| \le \|y\|$ with $N \le \|y\| \le N+1$ we have $|f(x,y)| \le M(N) \le b(N) \le B(y) \le A(x) + B(y)$, and similarly if
$\|y\| \le \|x\|$.
For the second case, note that $|f(x,y)| \le C(x) D(y)$ if $\log |f(x,y)| \le \log C(x) + \log D(y)$.
