Proving a claim regarding differentiability

In summary, the conversation discusses a function F(x,y,z) defined at point M_0(x_0,y_0,z_0) and its conditions, including being equal to 0 at M_0, having continuous partial derivatives, and the gradient of F at (x_0,y_0,z_0) being non-zero. It is also known that there is a function f(x,y) that satisfies F(x,y,f(x,y))=0 at M_0. The task is to prove that f(x,y) is not differentiable at (x_0,y_0).
  • #1
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Let F(x,y,z) be a function which is defined in the point M_0(x_0,y_0,z_0) and around it and the following conditions are satisfied:

1. F(x_0,y_0,z_0)=0
2. F has continuous partial derivatives in M_0 and around it
3. F'_z(x_0,y_0,z_0)=0
4. gradF at (x_0,y_0,z_0) != 0
5. It is known that there is a function f(x,y) so that F(x,y,f(x,y)) =0 in M_0 and around it

Prove that f(x,y) is differentiable in (x_0, y_0)
 
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  • #2
hmm sorry it seems that I have to prove
that f(x,y) is NOT differentiable in (x_0, y_0)
 

1. What is the definition of differentiability?

Differentiability is a mathematical concept that measures the smoothness of a function at a given point. A function is considered differentiable at a point if it has a well-defined tangent line at that point.

2. How can I prove that a function is differentiable?

To prove that a function is differentiable at a point, you can use the definition of differentiability and show that the limit of the difference quotient exists at that point. In other words, you need to show that the function has a well-defined derivative at that point.

3. What is the difference between continuity and differentiability?

Continuity and differentiability are related but distinct concepts. A function is continuous at a point if its value approaches the same value as the input approaches that point. Differentiability, on the other hand, requires not only continuity but also the existence of a well-defined tangent line at that point.

4. Can a function be differentiable at some points but not others?

Yes, it is possible for a function to be differentiable at some points but not others. For example, a function may not be differentiable at a point where it has a sharp corner or a discontinuity. However, it can still be differentiable at other points where the function is smooth.

5. Why is differentiability an important concept in mathematics?

Differentiability is a fundamental concept in mathematics because it allows us to study the behavior of functions at a given point. It is also essential in applications such as optimization, where we need to find the maximum or minimum of a function. Additionally, differentiability is a key component in the development of calculus and other advanced mathematical concepts.

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