Here, I have the function $u(x,y)$ =
\begin{cases}
\frac{x^{4/3}y^{5/3}}{x^{2} + y^{2}}, & \text{if $(x,y)\neq$ 0} \\
0, & \text{if $(x,y)$ = 0}
\end{cases}
I need to determine whether this function is continuous at $(0,0)$ and support my answer. I know how to prove it isn't continuous, by finding a limit of the first function which isn't equal to $0$, but I'm not sure how to prove that it is continuous. I feel like I should start by trying the epsilon delta proof, showing that
$\lvert f(x.y) – f(0,0)\rvert$ = $\lvert \frac{x^{4/3}y^{5/3}}{x^{2} + y^{2}}\rvert \le 1$, but I'm unsure how to do this precisely.
Best Answer
If $x=r\cos t,\ y=r \sin t,$ your function becomes (when $(x,y) \neq (0,0)$ so the polar is OK) $$\frac{r (\cos t)^{4/3} (\sin t)^{5/3}}{\cos^2 t+\sin^2 t}.$$ Then as $x,y \to 0,$ the denominator is always $1$ and the absolute value of the numerator bounded above by $|r|.$