You want to modify the functions f_{n}(x)=x^{n} (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(x^{2}),
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.
(In reply to
re: My Way by Tristan)
While it doesn't answer your exact question, the Wikipedia article on the subject Nonanalytic smooth function
does fairly adequately address your concerns. Using a graphing
calculator on exp(1/x) for 0<x<2, say, you willl see that the
graph practically coincides with the xaxis for 0<x<1/16, lending
credence to the assertion that at x=0, the function s is utterly
flat. A rigorous proof that the kth derivative of s at x=0 is
zero is somewhat involved, but depends on the fact that the kth
derivative of s for very small positive x always contains the factor
exp(1/x) so that the righthand derivative is zero just as the
lefthand derivative clearly is.
It is homework problem 1 of
paragraph 16 of chapter 4 of the wellknown book "Analysis on
Manifolds" by James R. Munkres to prove that s is of class Cinfinity
(has infinitely many, necessarily continuous, derivatives) on the real
line.
Edited on October 25, 2006, 11:59 pm

Posted by Richard
on 20061025 23:54:37 