For a continuous absolutely integrable function with absolutly integrable Fourier transform the result follows from:
by the translation theorem.
To prove you need a sequence of well behaved functions with known Fourier transforms , which as approximates the behaviour of a functional.
A suitable sequence can be constructed from Gaussians with decreasing spread parameters (who's FTs form a sequence of Gaussians with increasing spread parameters).
Then to prove you consider:
By construction the limit on the left hand side is , and on the right hand side you can exchange the limit and integral and so find it goes to right hand side of .
(Of course this is much more direct it you run amok with generalised functions or distributions, when you use instead of , and instead of )
(And also this is modulo the odd factor depending on how the FT has been defined)