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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?

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Real analytic is enough.

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

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