Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. The Fourier transform is defined as $\mathcal{F}(f) : \left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & \xi \mapsto \int_{-\infty}^{+\infty} f(x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x \end{aligned}\right.$. Show that $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.