MHB Continuity of a Function .... Conway, Definition 1.7.1 .... ....

[Math Amateur]

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ...

I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows:
View attachment 9497My question is as follows:

Is the above definition clear and correct? Is it usual?It seems to me Conway has defined continuity at any point $$x \in X$$ ... so why bother mentioning $$a$$ ... ?Can someone please clarify Conway's approach to continuity ...

Peter

 

[HallsofIvy]

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 1: The Real Numbers ... and in particular I am focused on Section 1.7: Continuous Functions ...

I need help with clarifying Definition 1.7.1 ...Definition 1.7.1 reads as follows:
My question is as follows:

Is the above definition clear and correct? Is it usual?It seems to me Conway has defined continuity at any point $$x \in X$$ ... so why bother mentioning $$a$$ ... ?Can someone please clarify Conway's approach to continuity ...

Peter

And HOW did he define "at any point [math]x\in X[/math]? As being continuous at every a\in X. That wouldn't make sense unless he had defined "continuous at a" first!

 

[Math Amateur]

And HOW did he define "at any point [math]x\in X[/math]? As being continuous at every a\in X. That wouldn't make sense unless he had defined "continuous at a" first!

Hi HallsofIvy ...

This is Conway's first and only definition of continuity for a real valued function of a real variable ...

I feel that there is a typo in it since as it stands it doesn't seem to make sense ...

Peter

 

[Opalg]

This is Conway's first and only definition of continuity for a real valued function of a real variable ...

I feel that there is a typo in it since as it stands it doesn't seem to make sense ...

Peter

You are quite correct, and once again you have found a typo in a respected textbook. Here, Conway claims to be defining continuity at $a$, but he actually defines continuity at $x$. In other words, he changes notation halfway through the definition.

Is the above definition clear and correct? Is it usual?

There are two ways of defining continuity. One is the $\varepsilon$-$ \delta$ definition, the other (used here by Conway) is to use sequences. Both definitions are "usual", and they are equivalent to each other. Each of them is useful in different contexts. Whichever of them is taken as the initial definition, most authors find it helpful to introduce the other one later, and to prove their equivalence.