# Differentiability in an open and closed intervals

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?

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

Staff Emeritus