Derivative of a derivative (2nd derivative) at only ONE point

[arestes]

Hi there!

Yesterday, while giving the definition of n times differentiable function (of a real function of a single variable, although a generalization of it to R^n or more sophisticated spaces would be nice to consider too, so if this makes sense more generally please also consider it) at a point (which involves the existence of all the lower order derivatives in a neighborhood and only the nth derivative at that point) someone asked my teacher in class if there are functions that have nth derivative (n>1) at only one isolated point while the lower order derivatives exist at an entire neighborhood of it. This seems like a natural question after getting the definition he gave.

At first my teacher thought the matter was easy to resolve by exhibiting the Dirichlet function times the identity which only has a derivative at zero but this cannot be the derivative of any function since it has 1st order discontinuities (finite jumps).

Does there exist a function that has a derivative in an entire interval and second derivative at only one of its points? more generally, a function with 1, 2,... n-1 th derivatives defined at (probably nested) intervals and nth derivative only at one point inside?


I've been trying to figure it out myself to no avail... My teacher gave up too but he thinks he's seen it a long time ago and had to do with some class of functions for which the fundamental theorem of calculus does not hold (?)... I wouldn't trust him much :Dcheers

 

[dharapmihir]

I think that derivative of an isolated point is not possible. We can have a derivative at a point only if we have a differentiable curve in its left and right neighborhood.

 

[arestes]

Hi Dharapmihir,

THERE ARE functions that are only differentiable in a single point and nowhere else (and they're even well defined for all the reals). The examples mentioned above are such functions. But because the functions that are derivatives of other functions cannot be completely arbitrary(they cannot have jump discontinuities), I cannot justify whether there exists or not a function that is the derivative of another one and is only differentiable at one point.

 

[Citan Uzuki]

It obviously suffices to find a continuous function differentiable at only one point, since all continuous functions have antiderivatives. Consider the function f(x) = x²w(x), where w(x) is the http://en.wikipedia.org/wiki/Weierstrass_function" . This is clearly continuous, it is differentiable at 0 via a squeeze theorem argument (w(x) is bounded, and x²→0 faster than x), but is not differentiable anywhere else (for if it were, w(x) = f(x)/x² would be differentiable there too).

 

[arestes]

Hi Citan Uzuki! thanks! this really works... I'll try to see if this generalizes to nth order too... If I can't figure it out myself I might come back and ask...
thanks again!

 

[arestes]

Of course, If $$\int^x_c t^2 w(t) dt$$ is differentiable at all of R then it is continuous and I can just keep on integrating and have a function which is only n times differentiable at a single point. I think I'm done with this, thanks!