Hello. I am confused about a point in complex analysis. In my book Complex Analysis by Gamelin, the definition for an analytic function is given as : a function f(z) is analytic on the open set U if f(z) is (complex) differentiable at each point of U and the complex derivative f'(z) is continuous on U p.45 Now, that is why I am shocked and confused to hear that there are functions that are infinitely differentiable but aren't analytic! It confuses me because we know that if a function is differentiable at a point p that means its continuous at that point. So if a function is inifinitely differentiable that means its continuous everywhere and its derivatives are continuous everywhere, right? Which is the definition of being analytic right? I have the same confusion for continuously differentiable functions which to my understanding are functions that are differentiable everywhere and their derivatives are continuous. Isn't that again all that is needed to show a function is analytic? Thanks I am so confused!