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S.B. Karavashkin

With it the mentioned particular forms of the function of a complex variable fully satisfy the conditions of continuous and one-valued mapping, if this last was understood not in Caushy - Riemann presentation, but more generally, in the Caushy or Heine sense.

Actually, "the function f ( z ) is continuous at the point z₀ if it was assigned in some vicinity z₀ (including the point z₀ itself) and

(4)

[1, p.20]. "At z z₀  the function w = f ( z ) has the number w₀ as its limit in the Caushy sense, if for each > 0 such  () > 0  exists that the inequality

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(5)

is true for all  z E C(, z₀)" [2, p.35]. Noting this, we can write (4) as

(6)

because the following statements are true:

Substituting any function from (3) into (6), we yield that in case of continuous functions of real arguments u and v, the function of complex variable w is continuous too. And vice versa, if at least one of functions of real arguments u and v was discontinuous, the function of complex variable w is discontinuous too, because at least one of equalities of (6) is violated.

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