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28 analysis/c.infinity.p




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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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