# Twice differentiable but not C^2

• mufq15
In summary, the conversation discusses finding an example of a function that is differentiable but not C^1 and extending this to C^2. The suggestion is made to consider the anti-derivative of the given function, but it may not be expressible in terms of elementary functions. The importance of the anti-derivative being equal or greater in differentiability is also mentioned. The conversation then explores the possibility of increasing the power of the amplitude to make the function even more differentiable. The example of f(x)=x^4sin(1/x) is discussed, with its derivative and second derivative also being mentioned. It is concluded that the second derivative of this function is not continuous.
mufq15
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.

Really?
What about an anti-derivative to the function you've posted?

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?

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")

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:
$$f(x)=x^{4}\sin(\frac{1}{x}), x\neq{0}, f(0)=0$$
Now, the DERIVATIVE of this function is readily found out to be:
$$f'(x)=4x^{3}\sin(\frac{1}{x})-x^{2}\cos(\frac{1}{x}), x\neq{0}, f'(0)=0$$
Now, see if you manage to differentiate this function everywhere, that is find the second derivative of f.

Oh, okay. The sin(1/x) in the second derivative makes it not continuous. Thanks for your help.

## 1. What does it mean for a function to be twice differentiable but not C^2?

A function is considered twice differentiable if its first and second derivatives exist. It is not C^2 if its second derivative is not continuous.

## 2. Can a function be C^2 but not twice differentiable?

No, a function cannot be C^2 if it is not twice differentiable. C^2 means that a function is twice differentiable and its second derivative is also continuous.

## 3. What is the significance of a function being twice differentiable but not C^2?

A function that is twice differentiable but not C^2 has a discontinuous second derivative. This means that the function has a sharp change in slope at some point, which can affect its behavior and make it more challenging to analyze.

## 4. How can I determine if a function is twice differentiable but not C^2?

To determine if a function is twice differentiable but not C^2, you can first find its first and second derivatives. Then, check if the second derivative is continuous at all points. If it is not, then the function is not C^2.

## 5. What are some real-life examples of functions that are twice differentiable but not C^2?

One example is the absolute value function, f(x) = |x|. Its first derivative is f'(x) = 1 for x > 0 and f'(x) = -1 for x < 0. Its second derivative is not continuous at x = 0, as f''(x) does not exist at that point. Another example is the function f(x) = x^(2/3), which has a discontinuous second derivative at x = 0.

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