Please define "Derivative" with reference to only one point.
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
He means the (second) derivative only exists at a single point. For the first derivative, I think an example would be the function that's 0 if x is rational and 1/x^2 if x is irrational, having a derivative only at 0. I can't think of an example where the second derivative exists at an isolated point.
Yeah, I know, it's kinda tough to find an example like that but for some reason I can't prove that they don't exist or even find a proof of nonexistence... The fact that the derivative of a function is a function defined on a smaller domain which cannot have first kind discontinuities messes up my attempts to find an example
It is known that there exist functions that are everywhere continuous but nowhere differentiable (Brownian motion, for example, or Weierstrass's function). Let be such a function, defined on the interval [–1,1]. Then the function is differentiable at the origin but nowhere else. It is a continuous function and is therefore the derivative of its integral (fundamental theorem of calculus). So the function has the properties that you are looking for.