# Differentiability in an open and closed intervals

• bubblewrap
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.

#### bubblewrap

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 :)

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?

Umm... I don't really understand, isn't that function twice differentiable in the closed interval [0,1]?

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##?

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]?