  We show that viscosity solutions to the normalized $p(x)$-Laplace equation
coincide with distributional weak solutions to the strong $p(x)$-Laplace
equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$
regularity for the viscosity solutions of the normalized $p(x)$-Laplace
equation. As an additional application, we prove a Rad\'o-type removability
theorem.
