Continuity of $f(x)=\frac{|x|\sqrt{x^4+y^2}}{|x|+|y|},$ when $(x,y)\neq 0$ and $0$ otherwise

Question

$f(x)=\begin{cases} \dfrac{|x|\sqrt{x^4+y^2}}{|x|+|y|},&\text{when $(x,y)\neq 0$}\\ 0,&\text{otherwise}\end{cases}$.

Is this function continuous? Also what are $f_x$ and $f_y$ at $(0,0)$?

I have no idea how to prove continuity. However I got $f_x=f_y=0$ at $(0,0)$.

Answer 1

HINT

For the missing answer, it suffices to note that

\begin{align*} \begin{cases} \displaystyle 0\leq \frac{|x|}{|x| + |y|} \leq 1\\\\ \displaystyle\lim_{(x,y)\rightarrow(0,0)}\sqrt{x^{4}+y^{2}} = 0 \end{cases} \end{align*}

then you apply the squeeze theorem.