Difference between definition of a limit if a function and definition of continuity.

Question

I am having trouble understanding why this particular difference exists between the definition of a limit of a function and definition of continuity.

Heres the definition of a limit of a sequence.

Let f : A → R, where A ⊂ R, and suppose that c ∈ R is an accumulation point of A. Then lim f(x) = L x→c if for every ε > 0 there exists a δ > 0 such that 0 < |x − c| < δ and x ∈ A implies that |f(x) − L| < ε.

I noticed that this definition requires $0 < |x - c| < \delta$

Let f : A → R, where A ⊂ R, and suppose that c ∈ A. Then f is continuous at c if for every ε > 0 there exists a δ > 0 such that |x−c| < δ and x ∈ A implies that |f(x)−f(c)| < ε.

whereas the definition of continuity above requires only $|x - c| < \delta$, so $|x - c$| can be 0.

Why is this the case?

Answer 1

Here are the differences:

$0 < |x - c| < \delta$ means that we are not imposing any conditions on the function for $x = c$, while in the definition of continuity, we have $|x -c | < \delta$. The difference is that in the definition of continuity at $c$, we insist that the function also be defined at $x = c$, but we don't in the definition of limit.

The conclusion $|f(x) - L| < \varepsilon$ in the definition of limit is different from $|f(x) - f(c)| < \varepsilon$ in the definition of continuity at $c$. The conclusion $|f(x) - f(c) | < \varepsilon$ actually is saying two things: that the limit exists, and further that the limit is $f(c)$.

To sum up, continuity at $c$ means essentially means the following three things:

1) The limit exists at $x = c$.

2) The function $f$ is defined when $x = c$.

3) The limit $f(x)$ as $x\to c$ is exactly $f(c)$.

A function having a limit need only satisfy the first of these three things.