Suppose that f Rn Rn and that a K, where K is a compact, connected subset of Rn Suppose further that for each x K there is a x 0 such that f(x) f(y) for all y Bx(x) Prove that f is constant on K that is, if a K, then f(x) f(a) for all x K ...

For each x K f is constant on B x x K Since K is compact and ...

Suppose that f: Rn Rn and that a K, where K is a compact, connected

Question:

Suppose that f: Rn → Rn and that a ∈ K, where K is a compact, connected subset of Rn. Suppose further that for each x ∈ K there is a δx > 0 such that f(x) = f(y) for all y ∈ Bδx(x). Prove that f is constant on K; that is, if a ∈ K, then f(x) = f(a) for all x ∈ K.