 Outward vector field zero theorem

A continuous vector field on a closed Euclidean ball whose boundary values have nonnegative outward component has a zero in the ball. Otherwise $T(x)=-Rg(x)/|g(x)|$ is a continuous self-map of the ball. Its fixed point, supplied by the Brouwer fixed-point theorem, lies on the boundary and satisfies $g(x)\cdot x=-R|g(x)|<0$, a contradiction. The argument turns a zero-finding problem into a fixed point problem without any differentiability assumption.