proof continuity of a multivariable function

Question

$$ f(x,y) = \begin{cases} \frac{x^3-xy^2}{x^2+y^2}, & \text{if }(x,y)\not= (0,0) \\ 0, & \text{if } (x,y)=(0,0) \end{cases} $$

how to prove $f(x,y)$ is continuous on $\mathbb{R}^2$? I know it must be continuous on $\mathbb{R}^2$, but according to the definition of continuity, how to use the $\delta- \varepsilon $way to prove?

I know how to prove the continuity at $(0,0)$ but want to know how to prove in the general case other than at $(0,0)$. I can't find a $\delta$ to fit the $\varepsilon$. I mean, let $|f(x,y)-f(x_1,y_1)|<\varepsilon$

$(x_1,y_1)$)is any point but not $(0,0)$.

and how to transform it to relate to a $\delta$ in order to have
$|(x,y)-(x_1,y_1)|<\delta$ then $|f(x,y)-f(x_1,y_1)|<\varepsilon$?

Answer 1

$x$ and $y$ are continuous functions. Moreover, the sum, product, and quotient (at points where the denominator is non-zero) of continuous functions are continuous.

If you really want a self-contained $\epsilon-\delta$ proof, you can use the proofs of the above lemmas and pump $\epsilon$ through the compositions. But that sounds like an awful lot of work for little gain...

Answer 2

You can argue by using the following inequality at $(0,0)$:

$$\left|\frac{x^3-xy^2}{x^2+y^2} \right|\leq |x|\frac{x^2}{x^2+y^2}+|x|\frac{y^2}{x^2+y^2}=|x|$$

Therefore, as $(x,y) \to (0,0)$ you have that $x \to 0$ and by the following inequality $f(x,y) \to 0=f(0,0)$.

In other points than $(0,0)$ the continuity is guaranteed because all the operations made are elementary and well defined.