In my answer here, I discussed the example of compactly supported continuous function $$g(x)= \begin{cases} \dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\ 0,&\text{otherwise} \end{cases}$$ which is not the Fourier transform of any $f\in L^1(\mathbb R)$, i.e. $g\not\in \mathcal F(L^1(\mathbb R))$. Looking through the proof, I think that one can smooth out $g$ at $x=\frac 12$, and get a function $\tilde g \in C_c(\mathbb R)$, supported on $[0,\frac 12]$, infinitely differentiable at EVERY point besides $x=0$ (though not even once differentiable at $x=0$), which is not in $ \mathcal F(L^1(\mathbb R))$.
I am now wondering if there is an example of compactly supported function $G\in C_c(\mathbb R)$ which is differentiable at every point, which is is still not in $ \mathcal F(L^1(\mathbb R))$
