Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $\mathcal{F}(f) \in \mathcal{S}$. Deduce that $f$ is of class $C^{\infty}$ on $\mathbb{R}$.