SELF

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S.B. Karavashkin and O.N. Karavashkina

With it, these particular forms of the function of complex variable fully satisfy the conditions of continuous and one-valued mapping, if we understand this last not in the Caushy - Riemann meaning but in more general meaning of Caushy either Heine.

Actually, “we call the function   f ( z ) continuous at the point z₀ , if it is determined at some vicinity z₀ (including the point z₀) and

(4)

[1, p.20].

Taking into account that “with  z z₀  the function w = f ( z )  has the number w₀ as its limit in the Caushy meaning if for any > 0   there exists such () > 0 that the inequality

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(5)

is true for all z E C(, z₀)” [2, p. 35], we may write (4) as

(6)

because the following statements are true:

Substituting any function from (3) to (6), we yield that in case of continuous functions of real argument u and v, the function of complex variable w is also continuous. And vice versa, if at least one of functions of real argument u and v is discontinuous, the function of complex variable w will be also discontinuous, as at least one equality of (6) will be violated.

The same simply we can prove the correlation between the one-valuedness of mapping of z onto w and that of the functions of real argument u and v.