I'm reading the definition of continuity at a point from Introduction to Real Analysis by Bartle-Sherbert text.
"Real Analysis" definition of continuity:
5.1.1 Definition Let $A \subseteq \mathbb{R}$, let $f: A \to \mathbb{R}$, and let $c \in A$. We say $f$ is continuous at$c$ if, given any number $\epsilon>0$, there exists $\delta > 0$, s.t. if $x$ is any point of $A$ satisfying $|x-c|<\delta$, then $|f(x)-f(c)|<\epsilon$.
If $f$ fails to be continuous at $c$, then we say $f$ is discontinuous at$c$.
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?