On some properties of finite sums of ridge functions defined on convex subsets of $\mathbb R^n$

Necessary conditions are established for the continuity of finite sums of ridge functions defined on convex subsets $E$ of the space $\mathbb R^n$. It is shown that under some constraints imposed on the summed functions $\varphi _i$, in the case when $E$ is open, the continuity of the sum implies the continuity of all $\varphi _i$. In the case when $E$ is a convex body with nonsmooth boundary, a logarithmic estimate is obtained for the growth of the functions $\varphi _i$ in the neighborhoods of the boundary points of their domains of definition. In addition, an example is constructed that demonstrates the accuracy of the estimate obtained.