We denote:

Let be given. We look for such that .

We take any such that and we are done.

Formally this definition is identical to the corresponding definition in Calculus. Thus we get easily the two following propositions:

- (i)
- is continuous at .
- (ii)
- is continuous at .
- (iii)
- If , then is continuous at .
- (iv)
- If , then is continuous at .

For a proof, we suggest to the reader to have a look at his/her course in Calculus. The needed adaptation is merely to understand the absolute value here as the absolute value of complex numbers instead of that of real numbers. The same remark applies to Proposition 1.5.

Recall that a polynomial function is a function of the form , where all the are given complex numbers and . We apply the two first properties of Proposition 1.4 to prove Corollary 1.6.

Recall now that a rational function is the quotient of two polynomial functions. Applying the last property of Proposition 1.4 to prove Corollary 1.6, we prove Corollary 1.7.

Noah Dana-Picard 2007-12-24