Hi all, I am having a little trouble understanding one of the concepts presented in my calculus class. I do not understand how the endpoints of an open interval can be differentiable. My teacher says that the endpoints of a closed interval can not be differentiable because the limit can not be approached from the left side of the left endpoint and the right side of the right endpoint. This makes sense to me, even though some research shows that there is no consensus on this subject. What I do not understand is why this argument can not be applied to open intervals as well. Wouldn't the endpoint of an open interval being differentiable imply that the left endpoint can be approached from the left? I do not understand how that is possible? Can anyone explain this concept to me more effectively? Thanks in advance for any help.
I'm pretty sure this has to do with the idea of finding the lowest number in the interval. If you specify some real number, there will always be a number less than that number that lies in the interval. The same kind of logic is used in the upper bound where some number will always exist to be greater but less than the upper bound. It has to do with the real numbers and the definition and properties of the set that defines your interval.
Your teacher is right: Endpoints of closed intervals are not differentiable, as the limit from one side does not exist. We don't ever have this problem with open intervals though. First note that open intervals don't have "endpoints" like closed intervals do, by definition, so the previous logic doesn't apply. For any point in an open interval, you can always mark off a little interval around it where that interval is still within the original open interval. This property makes open intervals also Open sets, a topological concept you may learn about later. 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. Whether they actually do approach limits, and if those left and right limits agree, is dependent on the function in question.
nonetheless it is often useful to define one sided derivatives, at the endpoints. E.g. when proving the fundamental theorem of calculus, one needs to check that the indefinite integral function satisfies the mean value theorem, which requires it to be continuous at the end points. One way to check (one sided) continuity at the endpoints is to check one sided differentiability.