  In this paper, we study the regularity of $\mathbb{R}$-differentiable
functions on open connected subsets of the scaled hypercomplex numbers $\left\{
\mathbb{H}_{t}\right\} _{t\in\mathbb{R}}$ by studying the kernels of suitable
differential operators $\left\{ \nabla_{t}\right\} _{t\in\mathbb{R}}$, up to
scales in the real field $\mathbb{R}$.
