Consider the function
$$f(x,y)=\begin{cases}|y/x^2|\cdot 2^{-|y/x^2|}, & x\neq 0\\ 0, & else\end{cases}$$
Show that $f$ isn't continuous at (0,0) while the restriction $f|_G$ on each line $G$ through the origin is continuous on $G$.
Solution:
So my idea was to just show that $f$ isn't continuous without restricting it at all. So by using polar coordinates, we get:
$\lim_{r\to 0} |\frac{1}{r}\frac{sin(\varphi)}{\cos(\varphi)^2}|2^{-|\frac{1}{r}\frac{sin(\varphi)}{\cos(\varphi)^2}|}=0$
since
$\lim_{k \to 0} \frac{k}{2^k}=0$
so $f$ is continuous at $(0,0)$.... so somehow what I just did isn't correct or isn't what I intended to. :/
Where's my mistake?