Calculus III - Multivariate Continuity

[Jonmundsson]

Homework Statement

Let
\begin{equation*}<br /> f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases}<br /> \end{equation*}

Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)?

Homework Equations

Polar coords

The Attempt at a Solution

If you convert f to polar it's easy to see that it is continuous (since it doesn't depend on \theta. I'm just wondering if the derivative of the polar function is dependent on \theta then it isn't continuous and therefore neither are f_x and f_y

Steps:

\displaystyle \lim _{(x,y) \to (0,0)} \dfrac{x^3 - y^3}{x^2 + y^2} = \lim _{r \to 0} \hspace{0.3em} r (cos^3 \theta - sin^3 \theta) = 0

Define g(r) = r (cos^3 \theta - sin^3 \theta) then g&#039;(r) = cos^3 \theta - sin^3 \theta and \displaystyle \lim _{r \to 0} cos^3 \theta - sin^3 \theta doesn't exist.

Cheers.

 

[SammyS]

Homework Statement

Let
\begin{equation*}<br /> f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases}<br /> \end{equation*}

Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)?

Homework Equations

Polar coords

The Attempt at a Solution

If you convert f to polar it's easy to see that it is continuous (since it doesn't depend on \theta. I'm just wondering if the derivative of the polar function is dependent on \theta then it isn't continuous and therefore neither are f_x and f_y

Steps:

\displaystyle \lim _{(x,y) \to (0,0)} \dfrac{x^3 - y^3}{x^2 + y^2} = \lim _{r \to 0} \hspace{0.3em} r (cos^3 \theta - sin^3 \theta) = 0

Define g(r) = r (cos^3 \theta - sin^3 \theta) then g&#039;(r) = cos^3 \theta - sin^3 \theta and \displaystyle \lim _{r \to 0} cos^3 \theta - sin^3 \theta doesn't exist.

Cheers.

How about ∂f/∂x and ∂f/∂y? Are the continuous at (0, 0) ?

 

[Jonmundsson]

They aren't. I'm asking whether it is a coincident that the derivative of g and the partials of f aren't continuous.