I have known that all the real analytic functions are infinitely differentiable.
On the other hand, I know that there exists a function that is infinitely differentiable but not real analytic. For example, $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is such a function.
However, the function above is such a strange function. I cannot see the distinction between infinitely differentiable function and real analytic function clearly or intuitively.
Can anyone explain more clearly about the distinction between the two classes?