I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (equivalent to holomorphicity). There are many important differences indeed analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.

For example in the complex case we have Liouville's theorem (any bounded complex analytic function defined on the whole complex plane is constant). But in the real analytic case this is false, just take $f(x)=\frac{1}{1+x^2}$ Actually, this example gives to us much more differences then this simple one.

In the real case it's a standard fact that any real-analytic function is $C^\infty$ but the converse is not true because of the following example \begin{equation*} {\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{x}}}&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}} \end{equation*}It is $C^\infty$ but it is not real-analytic at the origin. In light of this example I would like to ask this question

**How does the category of real-analytic manifolds differs from that of $C^\infty$ ones? Or rather, are there many examples of manifolds that admit smooth but not real-analytic atlases?**

## EDIT

Here I'll try to resume all the comments below. Thanks everybody for the answers!

(1) Take a look at this question Can every manifold be given an analytic structure? with much more discussion

(2) The main resut is that: *Any $C^1$ manifold admits a compatible real analytic structure (proved by Whitney)*

sureI've seen some closely related variant of this question before …. It's such a good and natural question that it's hard to imagine we went over a decade without asking it here! $\endgroup$6more comments