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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:

  1. $f(c)= \text{some real number}$
  2. $\lim_{x\to c} = \text{that same real number}$

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?

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You don't need to say "an $x$ value". Just say "at a point $c$" or something. –  dfeuer Aug 27 '13 at 22:59
    
@dfeuer ok, i see, thanks. But is all of this correct? the difference between continuity on closed intervals and continuity on open intervals? –  user437158 Aug 27 '13 at 23:05

1 Answer 1

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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:

  1. For every $\epsilon > 0$ there is a $\delta >0$ such that for all $x\in D$ such that $|x-c|<\delta$, $|f(x)-f(c)| < \epsilon$.

  2. 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.

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