Smooth function with infinite oscillation

calculus derivatives

The function

$$f(x) \; = \; \exp\left(-\frac{1}{x^2}\right)\cdot\sin\left(\frac{1}{x}\right)$$

with $f(0) = 0$ has the property you want. I believe Ulisse Dini, on p. 229 of his 1878 book Fondamenti per la Teorica delle Funzioni di Variabili Reali, was the first to publish such a function.

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