Continuous and tempered $\Rightarrow$ bounded by a polynomial?

Question

If $f$ is a continuous function that is moreover in the space of tempered distributions, is it true that $f$ is necessarily bounded by a polynomial function?

Answer 1

Implementing the idea of @zhw and at the same time answering the comment of @Goulifet:

Take a piecewise constant function $$f=\begin{cases}e^n,&x\in[n-e^{-2n},n+e^{-2n}]\quad n\in \Bbb N,\\0,&\text{otherwise}.\end{cases}$$ Clearly, it belongs to $L^1(\Bbb R)$, hence it is tempered.

Now make it continuous - instead of rectangular bumps use triangular ones of the same height over the same base. The function is still $L^1$ (hence tempered), continuous, but it is not bounded by a polynomial.