Let $f$ be defined as: $$ f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \end{cases} $$
What's the oscillation of $f$ at the point $(0,0)$?
Definition: oscillation of $f$ at the point $c$:
$$ O(f,c) = \inf_{c\in U}\sup_{x_1,x_2\in U}|f(x_1)-f(x_2)| $$, where $U$ is an open subset containing $c$.
$\lim_{x\rightarrow0}f(x,0) = 0 \neq 1/2 = \lim_{x \rightarrow 0}f(x,x)$
So, it's clear that $f$ isn't continuous at $(0,0)$, so $O(f,c) > 0$.
But I'm not really sure how to approach this. How to compute the values needed?
Making $y = \lambda x$ we have
$$ \frac{x y}{x^2+y^2} = \frac{\lambda}{1+\lambda^2} $$
and the oscillation is
$$ -\frac{1}{2} \le \lambda \le \frac{1}{2}$$
For
$$ \frac{\lambda}{1+\lambda^2} $$