Proving continuity of a function at a point - Homework $\Bbb R^2$ is using the Euclidean metric, $\Bbb R$ is using the standard $|y-x|$ metric.
We define $f:\Bbb R^2\rightarrow\Bbb R$ by
$$f(x,y) = \left\{\begin{array}{ll}
\frac{x^6+y^6}{x^2+y^2} & : (x,y)\neq(0,0)\\
0 & : (x,y)=(0,0)
\end{array}
\right.$$
I need to show this is continuous at $(0,0)$. While this is easy for me to understand in a more intuitive way, since $\frac{x^6+y^6}{x^2+y^2}$ would clearly have a discontinuity at $(0,0)$, I'm rubbish at proving these things using $\epsilon$ and $\delta$.

Take $x=rcos\theta,y=rsin\theta $
then $r^2=x^2+y^2$
So $|\frac{x^6+y^6}{x^2+y^2}|=|r^4(cos^6\theta+sin^6\theta)|\leq 2r^4=2(x^2+y^2)^2<\epsilon$
whenever $x^2<\frac{\epsilon}{4}$ and $y^2<\frac{\epsilon}{4}$
Taking $\delta =\sqrt\frac{\epsilon}{4}$ we get the required answer
