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78 - 79

S.B. Karavashkin and O.N. Karavashkina

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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 garrow.gif (842 bytes) z0  the function w = f ( z )  has the number w0 gequalitynon.gif (833 bytes) ginfinity.gif (844 bytes) as its limit in the Caushy meaning if for any gipsilon.gif (832 bytes) > 0   there exists such gdelta.gif (838 bytes)(gipsilon.gif (832 bytes)) > 0 that the inequality

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(5)

is true for all z gat.gif (834 bytes)E gcross.gif (839 bytes)C(gdelta.gif (838 bytes), 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.

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