Continuous function/x sin 1 over x/On 0,1/Infinitely many zeroes/Exercise

Prove that the function

${}f(x)={\begin{cases}x\sin {\frac {1}{x}}{\text{ for }}x\in {]0,1]},\\0{\text{ for }}x=0,\end{cases}}\,$

is continuous, and that it has infinitely many zeros.

Create a solution