Suppose that $f, \hat f\in L^1(\mathbb R^n)$. What smoothness on $f$ does this imply---i.e., what is the maximal $m$ such that $f$ is $m$ times differentiable (everywhere)? What is an example of a function $f$ such that $f,\hat f\in L^1(\mathbb R^n)$ is not $m+1$ times differentiable?
All I know is that $f$ has to be continuous. If $m=0$ is the maximal $m$, this means that there exists a function that is not differentiable but whose Fourier transform is in $L^1(\mathbb R^n)$. I'm having trouble finding such an explicit function $f$.