Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be a real analytic function which is non-constant. Are there simple conditions on $f$ to deduce that $$ \sup_{x \in \mathbb{R}^n} \|f(x)\|=\infty ? $$
Thoughts: From the analyticity alone, I can deduce that it is not compactly supported. However, that doesn't seem to be enough to guarantee that it is bounded in norm; i.e. consider $x\mapsto \exp(-x^2)$.
In constrast, it seems that all non-constant polynomial do satisfy this condition. So, the class of functions $\mathcal{X}$ which I'm interested in lyies somewhere between $$ \left\{f:\mathbb{R}^n\rightarrow \mathbb{R}^m:\mbox{non-constant polynomial}\right\} \subseteq \mathcal{X} \subset \left\{f:\mathbb{R}^n\rightarrow \mathbb{R}^m:\, \mbox{f is analytic}\right\}. $$
Is a characterization of $\mathcal{X}$ known?