All the generalized characteristics for the solution to a Hamilton–Jacobi equation with the initial data of the Takagi function

We determine all the generalized characteristics for the solution to a Hamilton–Jacobi equation with the initial data of the Takagi function, which is everywhere continuous and nowhere differentiable. This result clarifies how singularities of the solution propagate along generalized characteristics. Moreover it turns out that the Takagi function still keeps the validity of the recent results in Albano et al. (J. Differ. Equ. 268:1412–1426, 2020), in which locally Lipschitz continuous initial data are handled.