A complex variable *z* has a real part *x* and an imaginary part *y* and is written *z* = *x* + *iy*. Functions *f*(*z*) of a complex variable are our concern. Explicit examples are 1/*z*, *z*^{2}, and exp(*z*). Any function of a complex variable can be expressed as

where *u* and *v* are real functions.

A function *f*(*z*) is said to be continuous at *z* = *z*_{0} if given *any* positive number , one can find a number δ such that |*f*(*z*) − *f* (*z*_{0})| < provided that |*z* − *z*_{0}| < δ.

Let *z*_{0} = *x*_{0} + *iy*_{0}. It then follows that |*u* (*x*, *y*) − *u* (*x*_{0}, *y*_{0})| ≤ |*f*(*z*) − *f*(*z*_{0})| < if |*z* − *z*_{0}| < δ *i*.*e*., |*x* − *x*_{0}| < δ, *y* − *y*_{0}| < δ/. Hence, if *f*(*z*) is continuous, so is *u*(*x*, *y*) and similarly *v*(*x*, *y*). Conversely, it can be shown that if *u*(*x*, *y*) and *v*(*x*, *y*) are continuous, so ...

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