Let $f \in C(\mathbb{R}^n,\mathbb{R}^k)$. For most familiar such functions we can find some $\lambda>0$ such that $\lim_{\|x\|_{\mathbb{R}^n}to \infty} e^{-\lambda x} \|f(x)\|_{\mathbb{R}^k} = 0$. However, can we always find such a $\lambda$?
In other words, is a non-smooth equivalent of the schwartz space?
Let $n=k=1$ and let's define $f$ as $f(x)=\exp(x^2)$. Then we have that $$e^{-\lambda x}\left|e^{x^2}\right|=e^{x^2-\lambda x}$$ And it goes to $+\infty$ as $x \to \pm\infty$ no matter what $\lambda$ is.