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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 \(\displaystyle x \in X\) ... so why bother mentioning \(\displaystyle a\) ... ?Can someone please clarify Conway's approach to continuity ...

Peter

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 \(\displaystyle x \in X\) ... so why bother mentioning \(\displaystyle a\) ... ?Can someone please clarify Conway's approach to continuity ...

Peter