Problem of continuous function

Question

Define the function $g(x) = x^2\cos\frac1x$ for $x\ne 0$. What should be the value of $g(0)$ if $g(x)$ is a continuous function? Explain your work and justify your answer.

Frankly, I have no idea what I should do. I need some advices, thanks.

Answer 1

$|x^2\cdot cos\frac{1}{x}| \leq x^2$, thus by squeeze theorem, $g(x) \to 0$ as $x \to 0$. So you need to have $g(0) = 0$

Answer 2

Since $\cos \frac{1}{x}$ is bounded and $x^2$ tends to zero, $$ \lim_{x\to 0} x^2 \cos \frac{1}{x}=0. $$ So $g(0)$ must be $0$.