Riemann and Lebesgue integrals coincide?

Question

Is it enough to say that the function $f(x,y)=x^4+\sin(\frac{1}{y}),-1\leq x \leq 1,\, -1\leq y \leq 1$ is Riemann integrable because the set of its discontinuities has measure zero ($y=0$)? Then the Lebesgue integral would coincide with the Riemann one

Answer 1

Since the number of discontinuities of $sin(\frac{1}{y})$ is finite (there is only one discontinuity) the function is Riemann integrable over the interval.