Smooth function with infinite oscillation

Question

I'm curious to know if there's a function which:

Has an infinite number of solutions for $f(x)=0$ on $x \in [-a,a]$ for an $a > 0$ (like $f(x) = \sin(\frac 1 x)$)
Has an $n$th derivative for all $x\in[-a,a]$, for any $n$.
There is no interval $[b,c]\subset[-a,a]$ ($b<c$) for which $f(x)=0$ on the whole interval

Does such a function exist? If not, how could that be proven?

Well $f(x)\equiv 0$ fits the bill so far. – zhw. Mar 27 '17 at 21:31 — zhw., Mar 27 '17 at 21:31

score 6 · Accepted Answer · answered Mar 27 '17 at 21:56

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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