# one-sided continuity and one-sided derivative?

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?

• 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? – Emi Matro Aug 27 '13 at 23:05

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 $f(c)$.