Differentiability in an open and closed intervals

In summary, the conversation discusses the concept of differentiability in closed and open intervals, specifically the fact that a function can be differentiable more in an open interval than in a closed interval. This is due to the fact that a function may not be differentiable at the endpoints of a closed interval. An example is given with the function f(x)=x^(4/3) restricted to [0,1], which can be differentiated twice in the open interval (0,1) but only once at x=0. The conversation concludes with finding a function that is twice differentiable at both endpoints of a closed interval.
  • #1
bubblewrap
134
2
Is there an f(x) which is differentiable n times in a closed interval and (n+1) times in an open interval? I think I saw this in a paper related to Taylor's theorem (could be something else though). It didn't make sense to me, how can something be differentiable more in an interval that contains two less numbers? Is this something to do with the fact that a function might not be differentiable at the two end points of a closed interval because it does have the limit function coming from two sides? (a+0 and a-0) Sorry the post got long :)
 
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  • #2
Start with something simpler: ##f(x) = x^{4/3}## restricted to ##[0,1]##. This function can be differentiated twice on ##(0,1)##, but only once at x=0. Can you now find a function which does something similar at both end points?
 
  • #3
Umm... I don't really understand, isn't that function twice differentiable in the closed interval [0,1]?
 
  • #4
bubblewrap said:
Umm... I don't really understand, isn't that function twice differentiable in the closed interval [0,1]?

What is its second derivative? And can you evaluate it in ##0##?
 
  • #5
The second derivative of ##f(x)=x^4/3## is ##f(x)=4/9x^-2/3## right?
Oh is it because it can't be defined at ##x=0## that it's not differentiable at that point? Making it only twice differentiable in the open interval [0,1]?
 

What does it mean for a function to be differentiable in an open interval?

A function is said to be differentiable in an open interval if it has a derivative at every point within the interval. This means that the function is smooth and has a well-defined slope at every point within the interval.

How is differentiability in an open interval different from differentiability in a closed interval?

The main difference is that a function can be differentiable in an open interval but not in a closed interval. This is because in a closed interval, the function may not have a well-defined derivative at the endpoints. However, in an open interval, the endpoints are not included, so the function can still be differentiable at every point within the interval.

Can a function be differentiable at a point but not on an entire interval?

Yes, a function can be differentiable at a specific point but not on the entire interval. This can happen if the function has a sharp corner or a vertical tangent at that point, which would make the derivative undefined. However, the function can still be differentiable at all other points within the interval.

Is differentiability the same as continuity?

No, differentiability and continuity are two different concepts. A function can be continuous but not differentiable. This means that the function is smooth and has no breaks or jumps, but it may not have a well-defined derivative at every point. On the other hand, a function can be differentiable but not continuous. This means that the function has a well-defined derivative at every point, but it may have breaks or jumps in its graph.

What is the relationship between differentiability and smoothness?

Differentiability and smoothness are closely related, as a differentiable function is always smooth. However, a smooth function does not necessarily have to be differentiable. A function is considered smooth if it has continuous derivatives of all orders, while differentiability simply requires the existence of the first derivative.

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