# What is the difference between complex differentiable and holomorphic functions at a point?

What is the difference between:

a) a complex differentiable at a point $z$ or

b) a holomorphic at a point $z$,

for a function $f(z)$, on the complex plane?

It was my understanding that holomorphicity is just stronger than differentiability i.e. everywhere in the neighbourhood of that point is complex differentiable.

How do you go about showing where a complex function is either complex differentiable or holomorphic?

Any help would be greatly appreciated!

• You understood correctly – Paolo Leonetti Aug 17 '15 at 15:42
• What is an example of a function that is complex differentiable at a point, but isn't holomorphic at the same point? – spelufo Jun 28 '16 at 19:05
• Found one – spelufo Jun 28 '16 at 19:16

As a matter of fact, if a complex function is differentiable (i.e., $f'(z)$ exists and it is continuous for every $z\in\mathbb{C}$), then it is also analytic, that is, it is $C^\infty$ and admits a power series representation. This follows from the Cauchy integral formula, the cornerstone of complex analysis.
Note: It actually turns out that the continuity of $f'(z)$ is not necessary, but it actually follows from its existence. This is also called Goursat's lemma, which states that if $f(z)$ is differentiable in the complex sense, then $f'(z)$ is continuous.