We evaluate the descriptive set theoretic complexity of the space of continuous surjections from ${ℝ}^m$ to ${ℝ}^n$.

It is shown that a monotone function acting between euclidean spaces $R^n$ and $R^m$ is continuous almost everywhere with respect to the Lebesgue measure on $R^n$.

Let the spaces ${𝐑}^m$ and ${𝐑}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of ${𝐑}^m$, and let $f:A⟶{𝐑}^n$ be an order-preserving function. Suppose that $P$ is generating in ${𝐑}^m$, and that $Q$ contains no affine line. Then $f$ is locally bounded on the interior of $A$, and continuous almost everywhere with respect to the Lebesgue measure on ${𝐑}^m$. If in addition $P$ is a closed halfspace and if $A$ is connected, then $f$ is continuous if and only if the range $f\left(A\right)$ is connected.

We prove that if $\mathrm{grad}\left(f\right)\in L^p(R^n)$ for certain values of $p$, then$$\underset{x_1\to \infty }{lim}f(x_1,x_2,...,x_n)=\text{const.,}\text{a.e.}\text{in}{R}^{n-1}.$$