Show that $\lim_{K \to \infty} \sum_{k=-K}^{K} \hat{f}(k) e(k \alpha) = \dfrac{f(\alpha^+)+f(\alpha^-)}{2}.$

Question

I am studying Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan. The following is the beginning of Appendix D:

I could not understand the last sentence (the one contains D.2.). It's like a theorem that is well known in Fourier Analysis but neither search of the formula is possible in Internet nor could I find it in some books that I have and I search for the proof of this claim. I don't even know what do $\alpha^+$ and $\alpha^-$ mean? Could someone please let me know a source for a proof of the last sentence in the picture above and (D.2)? Or if it is easy to understand hints are also would be much appreciated.