Differentiability on an Open Interval

In summary, an open interval can be differentiable, which implies that the left endpoint can be approached from the left.
  • #1
ghelman
14
0
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.
 
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  • #2
ghelman said:
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
Ok, I understand now. Gib Z, you clarified things very well. Thank for the help guys.
 

What is differentiability on an open interval?

Differentiability on an open interval refers to the property of a function where there exists a derivative at every point within the given interval. This means that the function is smooth and continuous without any sharp or abrupt changes in its slope.

How is differentiability different from continuity?

Continuity refers to the property of a function where the output values change smoothly as the input values change. Differentiability takes this a step further by also requiring the existence of a derivative at every point within a given interval. In other words, a function can be continuous without being differentiable, but a function cannot be differentiable without being continuous.

What is the importance of differentiability on an open interval?

Differentiability on an open interval is important because it allows us to determine the rate of change of a function at any point within the interval. This is useful in many fields such as physics, economics, and engineering, where the rate of change of a quantity is often of interest.

What are the conditions for differentiability on an open interval?

In order for a function to be differentiable on an open interval, it must be continuous on that interval and have a defined derivative at every point within the interval. Additionally, the derivative must approach the same value from both the left and right sides at every point within the interval.

Can a function be differentiable at some points and not at others within an open interval?

Yes, it is possible for a function to be differentiable at some points and not at others within an open interval. This can happen when the function has a sharp corner or a vertical tangent at a certain point, making the derivative undefined at that point. However, if the function is continuous and has a defined derivative at every other point within the interval, it can still be considered differentiable on that interval.

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