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I've been wondering about something for a while now. As a computer scientist, I never had much in the way of analysis or advanced calculus, but there is something I have always assumed was true but have never been able to prove.

Say that you have a function f : R -> R which is infinitely differentiable (and continuous). The derivatives may eventually equal zero, but they must also be infinitely differentiable (and continuous) according to this same definition.

Further, assume that on some finite closed interval [a, b] the function is constant; that is, for any x in [a, b], f(x) = c for some finite value of c (with c constant).

Is it true that f(x) = c for all x in R?

Also, as a computer scientist, if the bottom line does not follow, I would definitely like an example of a function satisfying my requirements but which is not constant. No ad absurdum proofs, please. Thanks in advance for any help I may get in this matter.

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# Could use some help with a proof

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