Determining domain for C^1 function

AI Thread Summary
The discussion focuses on determining the differentiability of a function at the origin, specifically whether it is C^1. The partial derivatives with respect to x and y are given, but their limits do not exist as h approaches 0, indicating that the function is not C^1 at the origin. However, the function is C^1 at all other points due to the continuity of the partial derivatives, which are compositions of polynomials. The reasoning presented confirms that the function fails to be C^1 only at the origin. The issue with LaTeX formatting is acknowledged but does not detract from the main argument.
lys04
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Homework Statement
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Relevant Equations
Partial derivatives
The ####x partial derivative is equal to $$L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$ and the partial for ##y## is $$L \frac{4y}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$
Using the limit definition of partial derivatives I got the partial wrt ##x## is $$L \frac{h^{\frac{4}{5}}}{h}$$ which doesn’t exist as ##h## goes to ##0##. Similar argument for partial wrt ##y##. This means that ##f## isn’t ##C^1## at the origin, right?

At every other point the partial derivatives exist and is continuous because it’s a composition of a polynomial of two variables and ##x^2/5##, so ##f## is ##C^1## at all points except the origin.

Is the reasoning correct?
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I don’t think latex is working, not sure what’s wrong with it sorry
 
lys04 said:
I don’t think latex is working, not sure what’s wrong with it sorry
One problem is that { and } don't match.
 
Works very well with the additional }:

$$L \frac{4x}{5(x^{2}+y^{2})^{\frac{-3}{5}}}$$
 
\sqrt[2n+1]{x} is not differentiable at x = 0 for n \geq 1. This follows from the fact that x^{2n+1} has a point of inflection here.
 
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