$f(x , y)= \frac{x^3}{(x^2 + y^2)}$ if $(x , y) \neq (0 , 0)$. prove that f is continuous at any point Let $f(0 , 0) =0$  &
    $f(x , y)= \frac{x^3}{(x^2 + y^2)}$ if $(x , y) \neq (0 , 0)$.  prove that f is continuous at any point. The restriction of $f$ to any straight line is differentiable.  Can it be proved by not using sequential criteria?
Can anyone help me out?
I know that $|f(x , y)| < |x|$...That's how we can show that $f(x,y)$ is continuous at $(0,0)$. But how can I show that the function is continuous at other points as well.
 A: Let $(a,b) \in \mathbb{R^2}$
Take polar coordinates: $$x=a+r\cos{\theta}$$ $$y=b+r\sin{\theta}$$
Keep in mind that  $r \rightarrow 0 \Rightarrow (x,y) \rightarrow (a,b)$
and the final limit must be independent of $\theta$.
Lets compute it:
$$\lim_{r \rightarrow 0}f(a+r\cos{\theta},b+r\sin{\theta})=\lim_{r \rightarrow 0}=\frac{a^3+3r\cos{\theta}+3r^2 (\cos{\theta})^2+r^3 (\cos{\theta})^3}{a^2+b^2+2(ar\cos{\theta}+br\sin{\theta})+r^2}=\frac{a^3}{a^2+b^2}$$
The last limit does not depend on $\theta$.
A: You can prove the continuity at the origin by using polar coordinates and continuity on $\Bbb R^2-\{(0,0)\}$ is quite straight forward.
It is very handy to know the following results for multivariable functions, in fact it would be a good exercise to prove them.

(1) The sum, difference and product of two continuous functions is always continuous.
(2) The quotient of two continuous functions is continuous as long as the denominator is not $0$.
(3) Polynomial functions are continuous.
(4) Rational functions are continuous in their domains.
(5) If $f(x,y)$ is continuous and $g(x)$ is defined and continuous on the domain of $f$ then $g(f(x,y))$ is also continuous.
(6) Every projection map is continuous.

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The function $f$ in your question is a rational function on $\Bbb R^2-\{(0,0)\}$
Thus by $(4)$ we have $f $ is continuous on $\Bbb R^2-\{(0,0)\}$ and hence continuous on $\Bbb R^2$
A: Polar coordinates: 
$x = r\cos(\theta)$; $y= r\sin(\theta)$.
Define: 
$f(r=0, \theta)= 0$, $0\le \theta \lt 2π$
and
$f(r,\theta) = r(\cos(\theta))^3 $, 
for $0 \lt r$, $0 \le \theta \lt 2π$.
Consider:
$g(r):= r$, and $h(\theta) :=( \cos(\theta))^3$.
$g$ is continuous at any $r_0 \ge 0$.
$h(\theta)$ is continuous at any $\theta_0$ , $0 \le \theta_0 \lt 2π$.
At any point $(r_0,\theta_0)$:
$\lim_{r \rightarrow r_0} g(r)= r_0$, and 
$\lim_{\theta \rightarrow \theta_0} h(\theta) = h(\theta_0)$.
Combining: 
The product of $g$ and $ h$ , 
$f(r,\theta) = g(r)h(\theta)$ is continuous
at any point $(r_0,\theta_0)$, 
in particular at the origin:
$\lim_{r \rightarrow 0} \lim_{\theta \rightarrow \theta_0} f(r,\theta) = 0 = f(r=0, \theta_0)$
