Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^2\sin\!\left(\dfrac{\pi}{x^2}\right)$ for $x\neq 0$, $f(0)=0$. Then which of the following is TRUE?
A) $f(x)=0$ has infinitely many solutions in $\left[10^{-10},\infty\right)$
B) $f(x)=0$ has no solutions in $\left[\dfrac{1}{\pi},\infty\right)$
C) The set of solutions of $f(x)=0$ in $\left(0,10^{-10}\right)$ is finite
D) $f(x)=0$ has more than 25 solutions in $\left(\dfrac{1}{\pi^2},\dfrac{1}{\pi}\right)$