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My professor challenged me to determine an asymptotic expansion of this piecewise function :

$$f(x) = \left\{ \begin{array}{c} xsin(1/x) & if & x \in \mathbb{Q} & \\ \\ x & if & x \in \mathbb{R} \backslash \mathbb{Q} \\ \end{array} \right. $$

What I did immediately was to calculate the limit of x.sin(1/x) as x approaches 0 which is equal to 0, then I wrote f(x) = o(1) if x $\in$ $\mathbb{Q}$ since its limit is zero.

However, I couldn't continue solving the problem and I really can't figure out the next step.

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    $\begingroup$ The function is only continuous at $0$ and $\frac 2{(2k+1)\pi}$. It isn't asymptotic to anything nice. $\endgroup$ Feb 9 at 17:14

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