Prove that $\displaystyle f(x) = \left\{ \begin{array}{cc}x^4\tan \frac{\pi x}{2}, &\mbox{if}\ |x|<1\\x|x|, &\mbox{if}\ |x|\geq 1\end{array}\right.$ is an odd function.
We want $\displaystyle f(x)=-f(-x) \: \forall x \in \mathbb{R}$.
If $\displaystyle |x|<1, f(-x) = (-x)^4 \tan(-\pi x/2) = x^4 (-\tan (\pi x/2)) = -f(x)$
If $\displaystyle |x|>1, f(-x) = (-x)|(-x)| = \left\{ \begin{array}{cc}-x|-x|, &\mbox{if}\ x<0\\-x|x|, &\mbox{if}\ x>0\end{array}\right.
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so that in all cases $\displaystyle -f(-x)=f(x)$