### On Boman's theorem on partial regularity of mappings

Let $\Lambda \subset {\mathbb{R}}^{n}\times {\mathbb{R}}^{m}$ and $k$ be a positive integer. Let $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda $, the derivatives ${D}_{\xi}^{j}f\left(x\right):=\frac{{d}^{j}}{d{t}^{j}}f(x+t\xi ){|}_{t=0}$, $j=1,2,\cdots k$, exist and are continuous. In order to conclude that any such map $f$ is necessarily of class ${C}^{k}$ it is necessary and sufficient that $\Lambda $ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =({\eta}_{1},{\eta}_{2},\cdots ,{\eta}_{m})$ and homogeneous of degree $k$ in $\xi =({\xi}_{1},{\xi}_{2},\cdots ,{\xi}_{n})$. This generalizes a result of J. Boman for the case $k=1$. The statement and the proof of a theorem of Boman for the case $k=\infty $ is also extended...