Description
This article is from the Puzzles FAQ,
by Chris Cole chris@questrel.questrel.com and Matthew Daly
mwdaly@pobox.com with numerous contributions by others.
28 analysis/c.infinity.p
What function is zero at zero, strictly positive elsewhere, infinitely
differentiable at zero and has all zero derivatives at zero?
analysis/c.infinity.s
exp(-1/x^2)
There are infinitely many other such functions.
This tells us why Taylor Series are a more limited device than they might be.
We form a Taylor series by looking at the derivatives of a function at a given
point; but this example shows us that the derivatives at a point may tell us
almost nothing about its behavior away from that point.
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