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Practice problem

Real Analysis

By Victor Wang

Let f ⁣:R→Rf\colon \mathbb{R}\to \mathbb{R} be a function, smooth at every nonzero point x≠0x\neq0. Suppose f(x)=o(x3)f(x) = o(x^{3}) near the origin, meaning f(0)=0f(0) = 0 and lim⁡x→0f(x)x3=0.\lim_{x\to0} \frac{f(x)}{x^3} = 0. What is the best you can say about the smoothness of ff?

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Type a concept, like “slope” or “photosynthesis”.

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