<s>[INST] Define $f: \Bbb R^2 \to \Bbb R$ by $f(x,y) =\begin{cases} \frac{xy^2}{x^2 + y^4}, & \text{if $(x,y) \ne 0$} \\ 0, & \text{if $(x,y)= 0$} \end{cases}$ Define $f: \Bbb R^2 \to \Bbb R$ by
$f(x,y) =\begin{cases}
\frac{xy^2}{x^2 + y^4},  & \text{if $(x,y) \ne 0$} \\
0, & \text{if $(x,y)= 0$}
\end{cases}$
I'm to show that $f$ is continuous at any straight line THROUGH the origin, but not continuous AT the origin. I'm confused how this is possible. Is it a hole?
How would I do this?
 [/INST] To show that it's continuous along straight lines:
Approach the origin along the half line $(x,y) = (r\cos\theta,r\sin\theta)$, where $r>0$ and $r\to 0^+$.
\begin{eqnarray*}
\frac{xy^2}{x^2+y^2} &\rightsquigarrow& 
\frac{r^3\cos\theta\sin^2\theta}{r^2\cos^2\theta+r^2\sin^2\theta} \\ \\
&\equiv& \frac{r^3\cos\theta\sin^2\theta}{r^2} \\ \\
&\equiv& r\cos\theta\sin^2\theta
\end{eqnarray*}
As $r\to 0$, $r\cos\theta\sin^2\theta \to 0$ for all $\theta$.
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