Does there exist a function $u : \mathbb R^2 \rightarrow \mathbb R$ that is not continuous at $0 \in \mathbb R^2$, but whose restriction to every polynomial curve going through $0 \in \mathbb R^2$ is continuous? By a polynomial curve we mean the parameterized curve $(p(t), q(t))$ where $p$ and $q$ are polynomials.
I know there are functions that are separately continuous at $(0,0)$ for every polynomial line, but still, it is not joint continuous (continuous) but I don't really understand the reason and logic behind it