Differentiability in an open and closed intervals

  • #1
131
2

Main Question or Discussion Point

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 :)
 

Answers and Replies

  • #2
pwsnafu
Science Advisor
1,080
85
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
131
2
Umm... I don't really understand, isn't that function twice differentiable in the closed interval [0,1]?
 
  • #4
22,097
3,279
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
131
2
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]?
 

Related Threads for: Differentiability in an open and closed intervals

  • Last Post
Replies
4
Views
10K
Replies
11
Views
1K
Replies
6
Views
4K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
5
Views
2K
Replies
6
Views
6K
  • Last Post
Replies
7
Views
7K
Top