1. Periodic extension

The periodic extension will have value at 1 which is half way the value as you approach 1 from the left (which is 13) and as you approach one from the right (which is 1)

That average value is 7.

FSf(1) = 7 is the answer. Can someone check whether I am right or wrong.

2. Re: Periodic extension

Hey mathsforumhelp.

The value at discontinuities will be the average of the two values and is pointed out here:

Gibbs phenomenon - Wikipedia, the free encyclopedia

They specify this in terms of F(x) = [f(x-) + f(x+)]/2 where F(x) is the fourier series representation f(x-) = f(1-) = 13*1 = 13 and f(x+) = f(1+) = 1 so (13+1)/2 = 14/2 = 7.