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Thread: real analysis

  1. #1
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    real analysis

    Suppose $\displaystyle f$ is infinitely differentiable on $\displaystyle \mathbb{R}$.

    (a) If $\displaystyle \left|{f(x)\right|\leq 1 + \left|x\right|^m $ for some $\displaystyle m\in \mathbb {N},$ and for all $\displaystyle x\in\mathbb{R},$then $\displaystyle f(x)$ is a polynomial.
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  2. #2
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    This is not true: take $\displaystyle f(x)=\sin(x)$ and note the statement is true for any $\displaystyle m\in\mathbb{N}$.
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