What does it mean that you can "nudge" an open interval $(a,b)$ so that function is defined at $a$ and $b$?

Question

In this question on continuity, the OP says,

...an open interval would allow a left and right limit to exist since the limit can approach from both sides, correct?

and

...you can always mark off a little interval around it where that interval is still within the original open interval

Similarly in this answer to a (different) continuity question, the answerer mentions in parentheses in the comments,

because the function is defined beyond the open interval on which it is known to be continuous

The thing is I feel like I'm missing something crucial after reading these 2 statements. Somehow an open interval is extended a little more than its boundaries show, so that in the continuity test: $$\lim_{x\to c}f(x)=f(c)$$ At the very least, the RHS is defined and a function can be tested for continuity at $a$ or $b$. But surely that can't be right... right?

After reading this question of limits in open intervals and the wikipedia definition of continuous functions (which I am following), the one-sided limit at endpoints exist but $a,b\notin(a,b)$ means you don't deal with it since its outside the interval. This is consistent with wikipedia's statement:

In case of the domain $D$ being defined as an open interval ... the values of $f(a)$ and $f(b)$ do not matter for continuity on $D$

So exactly what was the OP and answerer referring to in that question and answer? And should one definitively ignore $f(a)$ and $f(b)$ when considering open intervals?

Answer 1

All this depends on the behavior of your function. If we define $f: (0,1) \to \mathbb{R}$ by $f(x) = \frac{1}{x}$ then $$ \lim_{x \to 1^-} f(x) = 1 $$ and so we could "extend" $f$ to a function $f_1 : (0,1] \to \mathbb{R}$ by letting $f_1(1)=1$. The function is now continuous (since $1$ is an endpoint you would only require the left-hand limit to exist, which it does). Of course, we could extend this even further to the right.

However, on the left end we have $$ \lim_{x \to 0^+} f(x) = +\infty $$ and so we cannot extend $f$ to have a value at $0$ so that the new extension is continuous.

Note that with my original $f$ it does not make sense to speak of $f(0)$ or $f(1)$ since they fall outside the domain. That is why I have given it the new name $f_1$ when extending it continuously on the right. That being said, the function $f_1$ is perfectly well-defined and continuous.

In short, you can't always "nudge" the interval, but you can if the limits exist. This depends on your function. For example, the function $\tan: \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to \mathbb{R}$ cannot be nudged at all.