Understanding the definition of continuity in real analysis

Question

I'm reading the definition of continuity at a point from Introduction to Real Analysis by Bartle-Sherbert text.

"Real Analysis" definition of continuity:

This seems a lot like the definition of "limits". Then I Google the definition of continuity and I see the following.

"Calculus" definition of continuity: $$ \lim_{x\to c} f(x) = f(c). $$

Ah yes, this is the definition I remembered from Calculus. Very simple.

So, I'm guessing the two definitions are equivalent, and that the reason I'm reminded of a limit in the Analysis definition is because that is exactly what it is.

Why does the Real Analysis text go so far just to avoid saying that continuity at a point is a limit?

I understand the need for precise definitions, especially in defining a first principle concept such as limit. But now that we already have a rigorous definition of limit, why not just state the definition in terms of a limit?

Answer 1

The two definitions are equivalent. If you expand out the definition of a limit into your second definition of continuity, then you get the first. That doesn't mean the first one is crappy or that the textbook is bad. It's just a difference in what the author chose to emphasize.

Answer 2

One simply need to read the book more carefully.

Bartle and Sherbert state clearly right after the definition you quoted that

As with the definition of limit, the definition of continuity at a point can be formulated very nicely in terms of neighborhoods. This is done in the next result. We leave the verification as an important exercise for the reader.

And right below Theorem 5.1.2, the authors make the following remark:

All of these are in the SAME page (page 125).

Reading carefully, slowly, and thoroughly helps.