Cauchy Condition for limit of a function to exist at a cluster point in the domain A professor of our college told us the following :

The necessary and sufficient condition that
$$\lim_{x \to a} f(x) = \ell$$  (where, $\ell$ is the limit)  is $$|f(x_{2})-f(x_{1})| < \epsilon, \text{ when } 0 < |x_{2}-x_{1}| < \delta.$$

Here the above is the Cauchy Criterion as told by my professor. But I was wondering whether this definition is correct or not because in the condition stated, there is no any mention of the cluster point $a$ in the domain where the $x$ converges to. And naturally the limit depends on the cluster point considered, then $a$ should have been somewhere in the Cauchy condition. Can anyone help me out pointing whether I am wrong or there is something missing in the if and only if condition written above.
 A: First of all, I assume your statement is:

The necessary and sufficient condition that
$$\lim_{x \to a} f(x) = \ell$$  (where, $\ell$ is the limit)  is: for all $\epsilon >0$, there is $\delta >0$ so that
$$|f(x_{2})-f(x_{1})| < \epsilon, \text{ when } 0 < |x_{2}-x_{1}| < \delta, x_1, x_2 \in A,$$
where $f$ is the domain of $A$.

As you pointed out, the equivalence is obviously false since there is no mention of $a\in A$ in the latter statement. Indeed, the latter (as stated in this answer) is the definition of uniform continuity of $f$, which implies in particular the continuity of $f$, so
$$ \lim_{x \to x_1}f(x) = f(x_1).$$
whenever $x_1$ is a cluster point of the domain $A$. So this is a much stronger statement then the existence of limit of $f$ at one point $a$.
One possible correct equivalence should be the following (I am not completely sure what your professor have in mind):

Theorem 1: Let $f: A\to \mathbb R$ be a function and $a$ is a cluster point of $A$. Then the following are equivalent:

*

*The limit of $f$ at $a$ exists.

*For all $\epsilon >0$, there is $\delta >0$ so that if $x_1, x_2\in A$ and $|x_1-a|<\delta$ and $|x_2 -a|<\delta$, then $|f(x_1) - f(x_2)|<\epsilon$.


The name "Cauchy-Criterion" suggests the above theorem. Indeed, in the case of sequence, we have the following

Theorem 2: Let $\{a_n\}$ be a sequence. Then the following are equivalent:

*

*Limit of $\{a_n\}$ exists as $n\to \infty$,

*$\{a_n\}$ is a Cauchy sequence: for all $\epsilon >0$, there is $N\in \mathbb N$ so that if $n, m\ge N$, then $|a_n - a_m|<\epsilon$.


The similarity is that they both characterize the existence of limit without any knowledge about the exact value of the limit.
