Suppose $f$ is a $C^\infty$ function from the reals to the reals that is never negative. Does it have a $C^\infty$ square root? Clearly the only problem points are those at which $f$ vanishes. 



Square root of a positive $C^\infty$ function.
The answer is "no". This is covered in great detail here:


<http://www.math.polytechnique.fr/~bony/BBCP_jfa.pdf>


