Can anyone provide an example of a realvalued function f with a convergent Taylor series that converges to a function that is not equal to f (not even locally)?
If you take the classic nonanalytic smooth function: $e^{1/t}$ for $t \gt 0$ and $0$ for $t \le 0$ then this has a Taylor series at $0$ which is, err, $0$. However, the function is nonzero for any positive number so it does not agree with its Taylor series in any neighbourhood of $0$. 


Another thing to note is that there are smooth functions whose Taylor series do not converge to the function in a neighborhood of ANY point! An easy example of this can be found here: 


I always thought the classic nonanalytic smooth function was exp(1/t^2) over the reals. This example is probably more satisfying to students (which is why you see it in texts) because when you look at that expression it's not obvious that anything funny should be happening at 0, whereas that's not so obvious for Andrew's piecewisedefined functions 

