A continuous function is continuous at an $x$ value (call the $x$ value that we're interested in $c$) if both of these conditions are met and are true:
- $f(c)= \text{some real number}$
- $\lim_{x\to c} = \text{that same real number}$
So, when we state this definition, we referring to a function being continuous on the open interval $(a,b)$, not the closed interval $[a,b]$, correct?
Because, an open interval would allow a left and right limit to exist since the limit can approach from both sides, correct?
Because for any point that is in an open interval, you can always mark off a little interval around it where that interval is still within the original open interval. So for any point in a given open interval, we have a little "space" on either side for our left and right limits to form.
But endpoints on $[a,b]$ cannot be approached from both sides, so a function defined on this interval is right-continuous at $a$ and left-continuous at $b$ and has only one-sided limits at endpoints $a$ and $b$?
So then a function on $[a,b]$ has only one-sided continuity, correct? Because how can an endpoint $a$ be approached from the left since it's an $endpoint$, it could only make the function right-continuous, not totally continuous.
And a derivative is usually defined on some differentiable interval $(a,b)$, but it could also be differentiable on a closed interval $[a,b]$, but in this case it would be a one-sided derivative, correct?
Why is continuity defined mostly on closed intervals, when closed intervals mean that it is only continuous from one-side, and open intervals mean that it's both right-continuous and left-continuous and hence has total continuity?
Best Answer
There are various ways to explain this, but probably the best way to start is to try to think of "continuous at a point" or "limit at a point" as being its own independent concept, rather than something defined in terms of right-continuity and left-continuity. In the case of the real numbers, there are a lot of ways to define this, but here are two good ones:
Let $D\subseteq \Bbb R$ and let $f\colon D\to \Bbb R$. Then $f$ is continuous at $c\in D$ iff:
For every $\epsilon > 0$ there is a $\delta >0$ such that for all $x\in D$ such that $|x-c|<\delta$, $|f(x)-f(c)| < \epsilon$.
Whenever $(x_i)$ is a sequence in $D$ that converges to $c$, $(f(x_i))$ converges to $f(c)$.
Edit:
Thus the notion of continuity at an endpoint is perfectly sensible. Also, it's quite reasonable to consider a derivative at an endpoint. I think you may be getting a little confused because there are so many theorems out there that apply when a function is continuous on an interval and differentiable in its interior—it's not that the function can't be differentiable there, but that the theorem doesn't need it to be.