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Complex Variable ... Differentiability Implies Continuity ... Conway and Mathews and Howell ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
I have been reading two books on complex analysis and my problem is that the two books give slightly different and possibly incompatible proofs that, for a function of a complex variable, differentiability implies continuity ...

The two books are as follows:

"Functions of a Complex Variable I" (Second Edition) ... by John B. Conway

"Complex Analysis for Mathematics and Engineering" by John H. Mathews and Russel W. Howell (M&H) [Fifth Edition] ...


Conway's proof that for a function of a complex variable, differentiability implies continuity ... reads as follows:



Conway - Proposition 2.2 .png



Mathews and Howell's proof that for a function of a complex variable, differentiability implies continuity ... reads as follows:



M&H - Theorem 3.1 ... .png



Now, as can be seen in the above proofs, Conway uses modulus/norm signs around the expressions in the proof while Mathews and Howell do not ...


Can someone explain the differences ... are both correct ... ?

Surely the Conway proof is more valid as the proof involves limits which involve ideas like "close to" which need modulus/norms ...


Hope someone can clarify this issue ...

Peter
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
They are equivalent. If [tex]\lim_{x\to a} f(x)= b[/tex] then [tex]\lim_{x\to a} |f(x)|= |b|[/tex] and if b= 0 the converse is also true.
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
They are equivalent. If [tex]\lim_{x\to a} f(x)= b[/tex] then [tex]\lim_{x\to a} |f(x)|= |b|[/tex] and if b= 0 the converse is also true.



Thanks for the help HallsofIvy ...

But ... it leaves me thinking that Conway made a pointless elaboration of his proof as modulus/norm signs were unnecessary ... indeed, I have no idea why he included them ...

Peter