How to prove that a complex function is differentiable at a point and where is it analytic in its domain I know the definition of complex differentiable functions and I know the Cauchy-Reimann equations that are used to prove that a function is differentiable in general, but my question is how to prove that a complex function is differentiable at a point?
Suppose $z_0 \in \mathbb{C}$.  We say that a function $f$ is complex differentiable at $z_0$ if $\lim_{z \to z_0} \frac{f(z) - f(z_0}{z - z_0}$ exists and is finite.  Implicit in this definition is the fact that complex differentiation is not defined at a single point $z_0$ but for all points in a open set $B(z_0, r)$ about $z_0$.  What I am saying is that a function cannot be complex differentiable at just one point $z_0$ without being complex differentiable in an open set (neighbourhood) about that point.  You stumbled upon a very good question that many students in complex analysis never think about.
Suppose $z_0 \in \mathbb{C}$.  We say that a function $f$ is complex differentiable at $z_0$ if $\lim_{z \to z_0} \frac{f(z) - f(z_0}{z - z_0}$ exists and is finite.  Implicit in this definition is the fact that complex differentiation is not defined at a single point $z_0$ but for all points in a open set $B(z_0, r)$ about $z_0$.  What I am saying is that a function cannot be complex differentiable at just one point $z_0$ without being complex differentiable in an open set (neighbourhood) about that point.  You stumbled upon a very good question that many students in complex analysis never think about.