<s>[INST] 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 $\delta$.
 [/INST] $x^6+y^6 \le (|x|^3+|y|^3)^2$.
Cauchy Schwarz gives
$|x|^3+|y|^3 = \langle (x^2,y^2), (|x|, |y|)\rangle \le \sqrt{x^4+y^4}\sqrt{x^2+y^2}$,  combining gives
$x^6+y^6 \le (x^4+y^4)(x^2+y^2)$.
If $0<x^2+y^2 \le 1$, then $x^4+y^4 \le x^2+y^2$, from which we get
${x^6+y^6 \over x^2+y^2} \le x^2+y^2$, from which
the result follows.
</s>