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We prove H\"older regularity of any continuous solution $u$ to a $1$-D scalarbalance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and theflux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. Forexample, $\ell = 3$ if $f(u) = u^3$. Moreover, we prove that at almost everypoint $(t,x)$, it holds $u(t,x+h) - u(t,x) = o(|h|^{\frac{1}{\ell}})$ as $h \to0$. Due to Lipschitz regularity along characteristics, this implies that atalmost every point $(t,x)$, it holds $u(t+k,x+h) - u(t,x) =o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply the results toprovide a new proof of the Rademacher theorem for intrinsic Lipschitz functionsin the first Heisenberg group.

H\"older regularity of continuous solutions to balance laws and applications in the Heisenberg group