We know that every differentiable function is analytic. In Conway's Functions of One Complex Variable (1973), states that every analytic function is infinitely differentiable and has a power series expansion about each point of its domain. So does this imply that every differentiable function is (1) infinitely differentiable and (2) has a power series expansion about each point of its domain?
Every differentiable function is infinitely differentiable?
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complex-analysis
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1Not all differentiable functions are analytic... – 2017-01-22
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3All **complex differentiable** (on their whole domain) functions are analytic. – 2017-01-22
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0Ok, but he said a differentiable function is analytic, I must have interpreted in a wrong context. – 2017-01-22
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0Ah, well we do have the complex analysis tag going – 2017-01-22
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0Yes. Every function with a complex derivative at all points of its domain (an open set) is infinitely differentiable and analytic on that domain. – 2017-01-22
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0Well, [holomorphic functions are analytic](https://en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions) and they have power series expansions. – 2017-01-22