- #1
mufq15
- 7
- 0
I need to find an example of such a function. I know that x^2sin(1/x) is differentiable but not C^1, but I'm having trouble extending this to C^2.
Twice differentiable but not C^2
- Thread starter mufq15
- Start date
-
- Tags
- Differentiable
Click For Summary
Homework Help Overview
The discussion revolves around finding an example of a function that is twice differentiable but not C^2. Participants explore the properties of differentiability and continuity in relation to specific functions, particularly focusing on the behavior of sine functions combined with polynomial terms.
Discussion Character
- Exploratory, Conceptual clarification, Mathematical reasoning
Approaches and Questions Raised
- Participants discuss the function x^2sin(1/x) and its differentiability, questioning how to extend this to a function that is twice differentiable but not C^2. There is consideration of using an anti-derivative and the implications of differentiability on the anti-derivative's continuity.
Discussion Status
The discussion is active, with participants sharing insights about the properties of differentiability and continuity. One participant suggests a specific function, f(x)=x^{4}sin(1/x), and encourages others to find its second derivative, indicating a productive line of inquiry. However, there is no explicit consensus on the example being sought.
Contextual Notes
Participants are navigating the complexities of differentiability and continuity, particularly in the context of functions that exhibit pathological behavior. There is an acknowledgment of the challenges in finding suitable examples and the limitations of expressing certain anti-derivatives in elementary terms.
- #2
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
- 10,165
- 138
Really?
What about an anti-derivative to the function you've posted?
What about an anti-derivative to the function you've posted?
- #3
mufq15
- 7
- 0
Um, I understand that that would work, but I don't think I know how to take an antiderivative of that (or am I just being silly?) Are you suggesting I just write it as the integral of that?
- #4
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
- 10,165
- 138
Yes, you won't be able to write your anti-derivative in terms of elementary functions, however the really important insight is that the anti-derivative of a function is always differentiable to a greater or equal extent as your original function. For functions finitely differentiable, the strict inequality holds, for infinitely differentiable functions, the "equality" holds.
Another choice should readily suggest itself by considering WHAT IS IT THAT MAKES YOUR FUNCTION ONCE DIFFERENTIABLE?
The answer is that the power of the x multiplied with the sine is big enough to kill off the crazy behaviour of the sine function!
(The amplitude of the function becomes so small that the pathological oscillation of the function becomes "irrelevant")
Thus, we may ask:
What if we make the power of the amplitude even bigger?
Might not this make also the derivative of our function not only continuous, but also differentiable?
Consider the function:
[tex]f(x)=x^{4}\sin(\frac{1}{x}), x\neq{0}, f(0)=0[/tex]
Now, the DERIVATIVE of this function is readily found out to be:
[tex]f'(x)=4x^{3}\sin(\frac{1}{x})-x^{2}\cos(\frac{1}{x}), x\neq{0}, f'(0)=0[/tex]
Now, see if you manage to differentiate this function everywhere, that is find the second derivative of f.
Another choice should readily suggest itself by considering WHAT IS IT THAT MAKES YOUR FUNCTION ONCE DIFFERENTIABLE?
The answer is that the power of the x multiplied with the sine is big enough to kill off the crazy behaviour of the sine function!
(The amplitude of the function becomes so small that the pathological oscillation of the function becomes "irrelevant")
Thus, we may ask:
What if we make the power of the amplitude even bigger?
Might not this make also the derivative of our function not only continuous, but also differentiable?
Consider the function:
[tex]f(x)=x^{4}\sin(\frac{1}{x}), x\neq{0}, f(0)=0[/tex]
Now, the DERIVATIVE of this function is readily found out to be:
[tex]f'(x)=4x^{3}\sin(\frac{1}{x})-x^{2}\cos(\frac{1}{x}), x\neq{0}, f'(0)=0[/tex]
Now, see if you manage to differentiate this function everywhere, that is find the second derivative of f.
- #5
mufq15
- 7
- 0
Oh, okay. The sin(1/x) in the second derivative makes it not continuous. Thanks for your help.
Similar threads
- · Replies 2 ·
- · Replies 2 ·
- · Replies 2 ·
- · Replies 1 ·
- · Replies 8 ·