Decide if a function is Riemann Integrable

Question

I have the following function: $f:[-\pi,\pi]\rightarrow [-\pi,\pi]\ \ f(x,y) = \left\{\begin{matrix} \frac{xy}{x^4+y^2}, & (x,y) \neq (0,0)\\ 0,& (x,y) = (0,0) \end{matrix}\right.$
I need to decide if the function is or it is not Riemann Integrable.
I am trying to do this by proving that the function is continuous. If it is it means that is Riemann Integrable too. And it is not because I found two paths for which the approach is different. The two paths are: $x = 0$ and $x = y$. Because of this, the function is not continuous and therefore is not Riemann Integrable, right?
I am not sure if this is the correct answare so if anyone could point me in the right direction it would be great. Thank you!

Answer 1

Because of this, the function is not continuous and therefore is not Riemann Integrable, right?

Not right. A discontinuous function can still Riemann integrable. For example, consider the function $f:[-1, 1] \to \Bbb R$ defined as $$f(x) = \begin{cases}1 & x \neq 0\\0 & x = 0\end{cases}.$$ This function is discontinuous at $0$ but still Riemann integrable on $[-1, 1]$.

Even in your question, the function is only discontinuous at one point and so, that would not matter.

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However, your function still is not Riemann integrable. This is because your function is not bounded. To see this, let $x = 1/n$ and $y = x^2$. Then, we have $$f(x, y) = \dfrac{x^3}{2x^4} = \dfrac{n}{2}.$$