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Let $f$ be a real non polynomial analytic function. Suppose that the function $f$ assumes arbitrarily large and arbitrarily small values, i.e., for all $K>0$, there are $a,b$ with $f(a)<−K$ and $f(b)>K$.

My question is:

Does this property persists for the derivatives $f^{(k)}, k=1,2,..$

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up vote 4 down vote accepted

Consider $f(x) = x + \sin(x)$.

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