  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.
