If a function is analytic, this means:

1, the function can be differentiated infinite times, and its Taylor series at a point “c” in its domain converges to the function it self for some x in (c-R, c+R). In other words, a analytic function can be written as

,  for all x satisfying

2, the quality mentioned in (1) must be applied to all “c” in the domain of f(x)

Here are some examples:

1, all absolute value functions are not analytic. For instance, f(x)=|x| is not analytic since at  x=0, it is no differentiable, which does not satisfy quality (1)

2, Any function that has some point a where is always true is not analytic except f(x)=0. Since all of these functions’ Taylor series at x=a must be   thus the series fails to converge to the function when  x values are not zero