Continuous derivative of the Fourier transform of a function

Define L_{\mathrm{bc}}^{1}\left(\mathbb{R}\right) to be the set of functions f:\mathbb{R}\rightarrow\mathbb{C} which are continuous, bounded, and satisfy
{\displaystyle \left\Vert f\right\Vert _{1}=\int_{-\infty}^{\infty}\left|f\right|<\infty}
Then given f\in L_{\mathrm{bc}}^{1}\left(\mathbb{R}\right) , define its Fourier transform \hat{f} to be
{\displaystyle \hat{f}\left(y\right)=\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi ixy}dx}
In my readings of the Fourier transform I encountered details about its derivative. In particular, if g\left(x\right)=-2\pi ixf\left(x\right) \in L_{\mathrm{bc}}^{1}\left(\mathbb{R}\right) , then \hat{f} is continuously differentiable and \hat{f}\,^{\prime}=\hat{g} . The proof I read of it left much to be desired, or at least much was left to the reader. I ended up putting considerable effort into a complete proof, using various resources from internet searches, and I thought I would give back with this proof itself. Link to proof: Continuous derivative of the Fourier transform of a function
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