Using epsilon and delta show the continuity of a function with two variables at the point (0, 0)

Question

The function is this : $f(x,y)=(x^3-y^2)\sin(x)$

I came to this : $ 0 < \sqrt{(x-0)^2+(y-0)^2} < \delta$ and $|(x^3-y^2)\sin(x)|$

But not sure where to go further..

Thanks in advance!

Answer 1

Can you show that $x^3-y^2$ and $\sin x$ are each continuous at $(0,0)$? Do you know how the proof goes that the product of two continuous functions is continuous? You can combine these arguments and essentially redo that proof to produce a direct $\epsilon$-$\delta$ proof of the continuity of $(x^3-y^2)\sin x$ at $(0,0)$.