Vector calculus - Prove a function is not differentiable at (0,0)

In summary, the conversation discusses the differentiability of the function ##f(x)## at ##(0,0)##. While it is shown that the function has partial derivatives and is continuous at this point, it is also proven that it is not differentiable. There was initially some confusion about the proof, but it was eventually understood that the use of vector coordinates was necessary to show the lack of differentiability.
  • #1
physics1000
104
4
##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}##
I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)##
but now I have to show if it differentiable or not at ##(0,0)##.
According to answers it is not and they proved by showing the vector coordinates at ##(1,1)## does not have a limit.
But I dont want a proof like that, I tried using definition and got stuck...
I know my Linear transformation is basically 0. but still got in trouble of the definition
 
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  • #2
Hi, is not needed anymore, at the end I managed to understand why they did vector coordinates, it is pratically trivial as I see it now
Thank thought :)
 

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields and their properties. It involves the use of vectors, which are quantities that have both magnitude and direction, to solve problems in calculus.

2. How do you prove a function is not differentiable at (0,0)?

To prove that a function is not differentiable at a specific point, you need to show that the limit of the difference quotient, which represents the slope of the tangent line, does not exist at that point. This can be done by evaluating the limit from different directions and showing that they give different results.

3. What does it mean for a function to be differentiable at a point?

A function is differentiable at a point if its derivative exists at that point. This means that the function has a well-defined tangent line at that point, and the slope of the tangent line can be calculated using the derivative of the function.

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

Yes, a function can be differentiable at some points but not others. This is because the existence of a derivative at a point depends on the behavior of the function near that point. If the function is continuous and has a well-defined tangent line at a point, then it is differentiable at that point.

5. What are some common examples of functions that are not differentiable at (0,0)?

Some common examples of functions that are not differentiable at (0,0) include absolute value functions, piecewise defined functions, and functions with sharp turns or corners at that point. These functions do not have a well-defined tangent line at (0,0) and thus are not differentiable at that point.

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