If f is infinitely differentiable and analytic on a dense set is f analytic?

In summary, the conversation discusses the question of whether a function f: R->R that is infinitely differentiable and analytic on a dense set is also analytic on the entire set. It is noted that a counterexample exists in the form of a bump function, which is analytic on \mathbb{R}\setminus \{0\} but not on the entire set. This is due to the fact that the series must approach different values from the left and right at the endpoints of the compact support.
  • #1
381
4
Let f: R->R. If f is infinitely differentiable and analytic on a dense set is f analytic? Is this true if we restric f to [0,1]?

note: by analytic I mean the radius of convergence of the taylor expansion is non-zero about every point.

Maybe this is simple but I was thinking about it and can't figure it out
 
Physics news on Phys.org
  • #2
Depending on what you want, the bump function

[tex]\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow \left\{\begin{array}{cc} e^{-1/x^2} & \text{if}~x\geq 0\\ 0 & \text{if}~x=0\end{array}\right.[/tex]

is a counterexample. It is analytic on [itex]\mathbb{R}\setminus \{0\}[/itex] but it is not analytic on entire [itex]\mathbb{R}[/itex].
 
  • #3
Right, any smooth function of compact support , say in [a,b], is not analytic at either of the endpoints a,b , because the series must approach different values from the left and right.
 
Last edited:

1. What does it mean for a function to be infinitely differentiable?

Infinitely differentiable means that a function has derivatives of all orders at every point in its domain. This includes the first derivative, second derivative, third derivative, and so on, as well as the limit of these derivatives as they approach infinity.

2. What is an analytic function?

An analytic function is a function that can be represented by a power series expansion. This means that the function can be expressed as the sum of a constant term, a linear term, a quadratic term, and so on, where each term is multiplied by a different power of the independent variable.

3. What is a dense set?

A dense set is a set of points that are closely packed together. In the context of this question, a dense set refers to a set of points in the domain of a function that are close enough together that the function is considered to be continuous at all points within that set.

4. Can a function be infinitely differentiable but not analytic on a dense set?

Yes, it is possible for a function to be infinitely differentiable but not analytic on a dense set. For example, the function f(x) = |x| is infinitely differentiable, but it is not analytic at x = 0. This is because the function is not differentiable at x = 0, so it cannot be represented by a power series expansion at that point.

5. Does a function being analytic on a dense set guarantee that it is analytic everywhere?

No, a function being analytic on a dense set does not guarantee that it is analytic everywhere. This is because there may be points in the domain of the function that are not included in the dense set and at which the function is not analytic. In order for a function to be analytic everywhere, it must be analytic at all points in its domain.

Suggested for: If f is infinitely differentiable and analytic on a dense set is f analytic?

Replies
5
Views
1K
Replies
1
Views
703
Replies
3
Views
1K
Replies
2
Views
916
Replies
3
Views
1K
Replies
10
Views
3K
Replies
1
Views
943
Replies
4
Views
154
Replies
1
Views
1K
Back
Top