Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux I

We discuss different notions of continuous solutions to the balance law \[u_t + (f(u ))_x =g \] with $g$ bounded, $f\in C^{2}$, extending previous works relative to the flux $f(u)=u^{2}$. We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kruzkov iso-entropy solutions, which yields uniqueness for the Cauchy problem. We also establish the ODE reduction on any characteristics under the sharp assumption that the set of inflection points of the flux $f$ is negligible. The correspondence of the source terms in the two settings is matter of a companion work, where we also provide counterexamples when the negligibility on inflection points fails.