Is there an $\mathbb{R}^2 \to \mathbb{R}$ function which does not have a limit at a point $a$ ($\nexists \lim\limits_{a} f$), but we get the same value as a possible value for the limit when we try to approach $f(a)$ through any line?
I think we would need $$f to not be differentiable around $a$. In that case, it would not be approximated by a linear function, so it could have a different limit with a non-linear path. However, I can't go further. And I don't even know if it's a goof approach.
This is not a homework, it was just mentioned by my calculus teacher in the previous semester.
Sure, let $A = \{(r \cos t , r \sin t) | t \in (0,2 \pi] , 0 \le r \le t\}$.
Define $f = 1_A$, then $f$ does not have a limit as $x \to 0$ but has the limit $1$ along each line through the origin.