A wheel with radius $1\text{ m}$ is rolled in a straight line through one complete revolution on a flat horizontal surface. How many meters did the center of the wheel travel horizontally from its starting location

Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k