Let $f \in L^1(\mathbb{R})$, where the Fourier transform of $f$ is defined by
$$\forall \xi \in \mathbb{R}, \quad \hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \mathrm{e}^{-\mathrm{i}x\xi} \,\mathrm{d}x$$
Show that, for every function $f \in L^1(\mathbb{R})$, $\hat{f}$ is defined and continuous on $\mathbb{R}$.