Understanding Differentiability and Continuity in Complex Analysis

[Math Amateur]

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:
View attachment 9258
Mathews and Howell's proof that for a function of a complex variable, differentiability implies continuity ... reads as follows:
View attachment 9259
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]

They are equivalent. If $$\lim_{x\to a} f(x)= b$$ then $$\lim_{x\to a} |f(x)|= |b|$$ and if b= 0 the converse is also true.

 

[Math Amateur]

They are equivalent. If $$\lim_{x\to a} f(x)= b$$ then $$\lim_{x\to a} |f(x)|= |b|$$ 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