Using your Head is Permitted

March 2010 solution

If one allows u to be zero then a simple example of one such f is f(x)=1/x.

The series Tf|a(x), in this case, is the sum from i=0 to infinity of a-i-1(x-a)i = (x/a-1)i/a.

This is a geometric series that converges when |x/a-1|<1, meaning when x ∈ (0,2a).

Therefore, for any d we can pick a=d/2, u=0.

This solution, however, uses u=0, and the riddle explicitly asks for a positive u. An example with a positive u would be

f(x)=e-1/(x-1) for x>1 and f(x)=0 for x≤1.

In this function, Tf|a(x) equals f(x) in the range (1,2a-1) for a>1. Therefore, we can choose a=d/2+1 for any d.

(My thank-yous to Christian Blatter for pointing out to me that u=0 violates the wording of the riddle.)

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