question about continuity: using polar coordinates Given a function $f\colon\mathbb R^2\rightarrow \mathbb R$ I want to study continuity. So I know the $\varepsilon-\delta$ and sequence criterion.
Now we had polar coordinates in lectures: set $x=r\cos\theta$ and $y=r\sin\theta$ and then consider $r\rightarrow 0$ for continuity in $(0,0)$.
This transformation seems very usefull for expressions like e.g. $\frac{xy^2}{x^2+y^2}$ but don't I approach the function only on all straight lines in $(0,0)^t$ ? And for continuity I have to approach $(0,0)^t$ however I want to which I don't do using polar coordinates. So why can I still use polar coordinates? Thanks for helping.

Add: I don't want a solution for the continuity of the example above. Clearly $|\frac{xy^2}{x^2+y^2}|\leq |y|<\epsilon$ and so continuity in $(0,0)$ with $f(0,0)=0$ which I also get with polar coordinates since $\cos^2\theta+\sin^2\theta=1$.

The use of polar coordinates and the limit $r\rightarrow 0$ to compute $\lim_{(x,y)\rightarrow (0,0)}f(x,y)$ is not a choice of path to approach your limit point.
With polar coordinates you are just "parametrizing" all the point in the neighborhood of $(0,0)$ through their distance from $(0,0)$ and their angle w.r.t. the positive $x$-axis.
 The limit $r\rightarrow 0$ is along any path connecting the starting point and  $(0,0)$, not (only) the straight lines. If such limit depends on the path itself, like in the case of the limit
$$\lim_{(x,y)\rightarrow (0,0)}\frac{xy}{x^2+y^2}$$
then we say that the function is not continuous at the limit point.
Of course, the use of polar coordinates is great for some functions, while it can be a bad choice for many others. To directly find a counterexample to continuity (if it exists!) is probably the best way to answer to such kind of problems.
