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I have known that all the real analytic functions are infinitely differentiable.

On the other hand, I know that there exists a function that is infinitely differentiable but not real analytic. For example, $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is such a function.

However, the function above is such a strange function. I cannot see the distinction between infinitely differentiable function and real analytic function clearly or intuitively.

Can anyone explain more clearly about the distinction between the two classes?

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    It is preferable to use `\tan`, `\sin`, `\log`, `\cos`, `\exp` etc in math mode instead of `tan`, `sin`, `log`, `cos`, `exp` etc2012-06-10
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    Back in May 2002 I posted a great deal of information at the Math Forum sci.math site (which for some reason never made it to google's sci.math site) about $C^{\infty}$ functions that fail to be analytic **at each point**. See [Part 1](http://mathforum.org/kb/message.jspa?messageID=387148) and [Part 2](http://mathforum.org/kb/message.jspa?messageID=387149). See also [Non-analytic function with convergent Taylor series everywhere](http://mathoverflow.net/questions/81613) at mathoverflow.2012-06-12

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