# Changes

## Uniformly bounded derivatives implies globally analytic

, 14:51, 7 July 2012
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==Statement==

===Global statement===
Suppose $f$ is an infinitely differentiable function on $\R$ such that, for any fixed $a,b \in \R$, there is a constant $C$ (possibly dependent on $a,b$) such that for all nonnegative integers $n$, we have:
Then, $f$ is a [[globally analytic function]]: the [[Taylor series]] of $f$ about any point in $\R$ converges to $f$. In particular, the Taylor series of $f$ about 0 converges to $f$.

==Examples==

The functions $\exp, \sin, \cos$ all fit this description.
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