A non-analytic smooth function is a real C∞-function that is not a real analytic function at one or more points. A standard example is
 |
(1)
|
Every derivative of 
at 0 is zero, so the Taylor series
of 
about 0 is identically zero. However, 
for 
, and hence the Taylor series
does not equal 
on any neighborhood of 0.
Thus a function can be differentiable to every
order without being a real analytic function.
