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The function f(z) is said to be analytic at z∘ if its derivative exists at each point z in some neighborhood of z∘, and the function is said to be differentiable if its derivative exist at each point in its domain. So whats the difference?

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    I think you mean "differentiable" not "differential." – 2017-02-08
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    Thanks for the correction. – 2017-02-08
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    The terms each have their own meaning -- but it turns out that the complex derivative has the surprising property that in this special case, they turn out to be equivalent. – 2017-02-08
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    You forgot "entire" which also falls in this question, so now we have 4 (!!) quasi synonymous terms. – 2017-02-08

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Holomorphic functions are complex functions, analytic functions are not necessarily complex.

Both, holomorphic and analytic functions, are infinitely continuous differentiable. But a differentiable functions is not necessarily infinitely differentiable, moreover: an infinitely differentiable function is not necessarily analytic or holomorphic.

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    This means that analytic functions are complex and real functions, Holomorphic functions are complex, and differentiable functions are real functions? :) – 2017-02-08
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    @user2129579 No, differentiable functions can be real or complex. – 2017-02-08
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    Ok. Can I say that differentiable functions are finitely or infinitely differentiable, and analytic functions are always infinitely differentiable? – 2017-02-08
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    @user2129579 yes but I recommend that you take a book of analysis to learn this topic properly. – 2017-02-08
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Take $f(z) = z^2 \cdot 1_{\mathbb Q}$. This function is not holomorphic but it is differentiable at $0$.

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    $1_{\mathbb Q}(z) = 1$ if $z \in \mathbb Q$ and $0$ else. – 2017-02-08
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    (Another comment : holomorphic = analytic, but this is not an obvious theorem. This is normally done in a first course in complex analysis.) – 2017-02-08