# Continuous and differentiability

• MHB
• Joe20
In summary, the question asks for advice on the correctness of the steps taken in finding the derivative of f at x=0. The conversation also discusses how to show the existence of f''(0). The steps taken so far show that f is differentiable at 0 and f'(0)=0, and for x ≠ 0, f'(x) = 2x for x>0 and f'(x) = -2x for x<0. The final decision on whether f''(0) exists will depend on further calculations or using the definition of a derivative or derivative tests.
Joe20
Hello,

I have attached the question and the steps worked out. I am not sure if my steps are correctly. Need advise on that.
Next, I am not sure how to show f''(0) exist or not. Thanks in advance!

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You have shown that $f$ is differentiable at $0$ and that $f'(0) = 0$. For $x\ne0$ you can use the usual rules for differentiation, to see that $f'(x) = 2x$ if $x>0$ and $f'(x) = -2x$ if $x<0$. Therefore $f'(x) = \begin{cases}2x&\text{when }x\geqslant0,\\-2x&\text{when }x<0.\end{cases}$ Now you have to decide whether that function is differentiable at $x=0$.

Hi there,

Thank you for sharing your question and steps. It's always helpful to see the work that has been done so far. As for whether your steps are correct, I would suggest double-checking your calculations and making sure you have followed all the necessary rules and formulas. You could also ask a classmate or your teacher for feedback to ensure accuracy.

Regarding f''(0), one way to show its existence is by using the definition of a derivative. You could show that the limit of the difference quotient as x approaches 0 exists, which would prove the existence of the second derivative at 0. Alternatively, you could also check for differentiability at 0 using the first and second derivative tests.

I hope this helps. Good luck with your problem!

## 1. What is the difference between continuity and differentiability?

Continuity refers to a function's ability to be drawn without any breaks or gaps. It means that the values of the function remain close together as the input values change. Differentiability, on the other hand, refers to a function's ability to have a well-defined derivative at a specific point. In other words, it is the smoothness of a function at a particular point.

## 2. How do you determine if a function is continuous?

A function is continuous if it satisfies three conditions: the function is defined at a point, the limit of the function exists at that point, and the limit and the function value at that point are equal. In other words, the left and right-hand limits of the function must equal the function value at that point.

## 3. What is the difference between continuity and uniform continuity?

Continuity refers to a function's behavior at a specific point, while uniform continuity refers to a function's behavior over an entire interval. A function is uniformly continuous if the difference between two input values does not affect the difference between their corresponding output values. This means that the function's rate of change is consistent throughout the interval.

## 4. How do you determine if a function is differentiable?

A function is differentiable if its derivative exists at a specific point. The derivative of a function is the slope of the tangent line at that point. To determine if a function is differentiable, you can use the limit definition of the derivative or check if the function satisfies the differentiability conditions.

## 5. What is the relationship between continuity and differentiability?

Continuity is a necessary condition for differentiability. If a function is differentiable at a point, it must also be continuous at that point. However, a continuous function may not necessarily be differentiable. This can happen when there is a sharp turn or corner in the graph of the function, which would cause the derivative to be undefined at that point.

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