[Math] prove that $f$ continuous at $(x)=0$ using a $\epsilon$-$\delta$ proof

Question

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$-$\delta$ proof
$$
f(x, y) = \begin{cases}
x^2sin(\frac1x),&x\neq 0
\\
0,&x = 0
\end{cases}
$$

Answer 1

If $0<|x|<\delta$ then $$|f(x)-f(0)| = |x^2 \sin(1/x)|\le|x|^2 < \delta^2$$ Thus we for every $\epsilon>0$ we can choose $\delta = \sqrt{\epsilon}$ such that...