Do there exist bump functions with uniformly bounded derivatives? [duplicate]

real-analysis functional-analysis

Solution 1:

If that bound held, then $\phi$ would be equal to its Taylor series everywhere (by considering the LaGrange form of remainder in Taylor's theorem), no matter where we center the series. In particular, we could center the series at a point where all derivatives are $0.$ That would give $\phi \equiv 0.$

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