You want to modify the functions f_n(x)=xⁿ (n = 1, 2, 3, ...) in the following way: Outside the interval [-1,1] the functions should remain completely unchanged. At zero, the new functions should be 1 (instead of 0). And the most important requirement: The functions should remain continuously differentiable infinitely many times, everywhere.

You can achieve this by adding a "hump" function to f_n. But not any function will do: For instance, you cannot add a Gaussian exp(-x²), because that would change f_n outside the interval [-1,1]. You also cannot add a sine hump, (0.5 + 0.5*sin(π x)) for x in [-1,1] and 0 otherwise, because the second derivative would become discontinuous at -1 and 1. Find a hump function which will do the trick.

Can you prove that the function s(x) is infinitely smooth?  It just doesn't seem obvious to me.

Posted by Tristan on 2006-10-25 21:06:24