A function property to guarantee that being constant on an interval implies identically constant

Question

Let $f:\mathbb R\rightarrow \mathbb R$. Suppose we know that $f $ is a constant on some open/closed interval. Which condition does guarantee that $f $ is constant on $\mathbb R$?

Clearly, continuity is not enough. Differentiable? $C^1$? smooth? real analytic?

score 1 · Answer 1 · answered Nov 10 '14 at 05:54

1

Real analytic is enough.

Smooth is not enough. See http://en.wikipedia.org/wiki/Bump_function

answered Nov 10 '14 at 05:54

orangeskid

56,630

can u please prove this fact that a real analytic function satisfies the problem posed by OP – Learnmore Nov 10 '14 at 06:02
@learnmore See The Identity Theorem for real analytic functions – Nov 10 '14 at 06:12

A function property to guarantee that being constant on an interval implies identically constant

1 Answers1