In order for a complex function of a  single complex variable to be differentiable it must be differentiable both parallell to the imaginary axis δy →0 and parallell to the real axis δx →0.

This condition leads to the CAuchy –Riemann equations-

The Cauchy–Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are the two equations:

(1a)    

and

(1b)    

Proof:

Suppose that

is a function of a complex number z. Then the complex derivative of ƒ at a point z₀ is defined by

provided this limit exists.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis; in either case it should give the same result. Approaching along the real axis, one finds

On the other hand, approaching along the imaginary axis,

The equality of the derivative of ƒ taken along the two axes is

Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of (this is called a domain in ). 

 function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. This is not true for real differentiable functions.