Let $\displaystyle f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a function such that along every line $\displaystyle y=Ax$ and the line $\displaystyle x=0$, the function is continuous at $\displaystyle (0,0)$. By this I mean if we let $\displaystyle L_{\theta}$ be one of the lines in the x-y plane given by rotating an angle $\displaystyle \theta$ anti-clockwise from the x-axis, and let $\displaystyle \{ a_n \}$ be a sequence converging to $\displaystyle (0,0)$ s.t. $\displaystyle a_n\in L_{\theta}$ for all $\displaystyle n$, then $\displaystyle f(a_n) \rightarrow f(0,0)$ as $\displaystyle n \rightarrow \infty$.

Is it true then that $\displaystyle f$ is continuous at $\displaystyle (0,0)$? So if we take an arbitrary sequence converging to $\displaystyle (0,0)$, say $\displaystyle \{ b_n \}$, then does $\displaystyle f(b_n) \rightarrow f(0,0)$? I was thinking that the each element $\displaystyle b_n$ lies on some line$\displaystyle L_{\theta_n}$ so that there are countably many lines that the sequence lies on, but I'm not sure if that means anything. Is the statement even true?

Thanks for any help, just something that was bothering me