Investigating the Differentiability of the Function ( f(x, y) begin{cases} frac{x^4 y^4}{x^2 y^2} text{if } xy eq 0 0 text{if } xy 0 end{cases} )

In this article, we will delve into the investigation of the differentiability of the function ( f(x, y) begin{cases} frac{x^4 y^4}{x^2 y^2} text{if } xy eq 0 0 text{if } xy 0 end{cases} ) in the 2-dimensional space ( mathbb{R}^2 ).

The function is continuous and differentiable in ( mathbb{R}^2 setminus {(0, 0)} ). The primary concern is to verify the continuity and differentiability at the point ((0, 0)).

Continuity at ((0, 0))

To determine the continuity at the origin, we need to evaluate the limit of ( f(x, y) ) as ((x, y) to (0, 0)).

[lim_{(x, y) to (0, 0)} f(x, y) lim_{(x, y) to (0, 0)} frac{x^4 y^4}{x^2 y^2}]

Using polar coordinates ( x r cos theta ) and ( y r sin theta ), where ( r to 0 ) and ( theta ) is an angle, we can rewrite the function as:

[lim_{r to 0} frac{(r cos theta)^4 (r sin theta)^4}{(r cos theta)^2 (r sin theta)^2} lim_{r to 0} frac{r^8 cos^4 theta sin^4 theta}{r^4} lim_{r to 0} r^4 cos^4 theta sin^4 theta 0]

Since this limit is zero for all values of ( theta ), the function is continuous at ((0, 0)).

Differentiability at ((0, 0))

To establish differentiability at ((0, 0)), we need to calculate the partial derivatives and their continuity. We start by defining the function ( f(x, y) ) in terms of ( x ) and ( y ). For ((x, y) eq (0, 0)), we have:

[f_x frac{2x^5 - 2x^3 y^2 - xy^4}{x^4 y^2}] [f_y frac{2x^4 y - 2x^2 y^3 - y^5}{x^4 y^2}]

The partial derivatives are defined and continuous in ( mathbb{R}^2 setminus {(0, 0)} ). Next, we need to check their continuity at ((0, 0)).

Continuity of ( f_x ) at ((0, 0))

[lim_{(x, y) to (0, 0)} f_x lim_{(x, y) to (0, 0)} frac{2x^5 - 2x^3 y^2 - xy^4}{x^4 y^2}]

Using polar coordinates, we get:

[lim_{r to 0} frac{2r^5 cos^5 theta - 2r^5 cos^3 theta sin^2 theta - r^5 cos theta sin^4 theta}{r^6 cos^2 theta} lim_{r to 0} r left( 2 cos^5 theta - 2 cos^3 theta sin^2 theta - cos theta sin^4 theta right) 0]

Since the trigonometric terms are bounded, the limit exists and is equal to zero. Thus, ( f_x ) is continuous at ((0, 0)).

Continuity of ( f_y ) in ( mathbb{R}^2 )

For ( f_y ), the same approach as for ( f_x ) can be used to show that it is continuous in ( mathbb{R}^2 ).

Since the partial derivatives are continuous and the function is continuous, we can conclude that the function ( f(x, y) ) is differentiable at ((0, 0)).

Conclusion

The function ( f(x, y) begin{cases} frac{x^4 y^4}{x^2 y^2} text{if } xy eq 0 0 text{if } xy 0 end{cases} ) is differentiable at any point in the 2D space ( mathbb{R}^2 ).