Unnecessary use of left and right limits?

Question

In many texts, I have seen the following definition of continuity at a closed interval:

Let $f: [a,b] \to \mathbb{R}$. Let $x \in [a,b]$. Then $f$ is continuous at $x \in (a,b)$ if $\lim_{y \to x}f(y) = f(x)$ and for $x = a$, if $\lim_{y \to a+} f(y) = f(a)$ (similarly for $b$).

I really don't see the need in distinguishing the endpoints in the definition, because $\lim_{x \to a} f(x)$ is equal to the right limit, provided one of the two exists.

Can someone clarify?

Answer 1

You are right. At a left endpoint, by definition, the "two-sided" limit is the same as the right limit, because only arguments inside the domain are considered.

Answer 2

Basically, we are saying 'continuous on an interval' here, we care about how continuous it is of the interval itself, not of anything outside of it, that is, we don't care what the left or right outside of the interval say. This comes with a problem at the endpoints as it is very possible for something to be continuous from the direction of the interval, and 'match' our ideas of how continuity should work(this is more formally the left or right handed limit depending on which endpoint you are at), yet in general be discontinuous when we approach it from outside the interval(that is, taking the other limit from the otherside). To solve this, we make our definition match this idea by taking at the endpoints limiting only from the direction in which the interval exists.

This is a far more useful definition when we consider analyzing functions only over this interval.

So, basically, it is possible for intervals to 'act' continuously despite being discontinuous in general at the endpoints.

Just because the limit $\lim_{x \to a} f(x)$ existing means $\lim_{x \to a+} f(x)$, doesn't mean $\lim_{x \to a+} f(x)$ means $\lim_{x \to a} f(x)$ exists, and so, with our above concept of continuity on an interval, do need to specify the direction of the limit.

To see an example of how a continuous interval can exist with discontinuous endpoints, an image below is posted.

The function is, in general, discontinuous, but behaves 'continuosuly' in the interval [-1,2] and so match our definition to this 'intuition'.