Examine the continuity and discontinuity of the following function at $(0,0)$
$$f(x,y)=\begin{cases}{x^3\cos({1\over y})+y^3\sin({1\over x})\over {x^2+y^2}} & x\neq 0\neq y\\ 0 & \text{otherwise}\end{cases}$$
I tried to prove continuity using the definition of limits
$$|f(x,y)-f(0,0)|=|{x^3\cos({1\over y})\over x^2+y^2}|+|{y^3\cos({1\over x})\over x^2+y^2}|$$
$$\le|{x^3+y^3\over x^2+y^2}|$$
what do i do next ? Need some hint on proving differentiability as well .
Best Answer
Hint: $$\left|\frac{x^3}{x^2+y^2}\right| = |x|\left|\frac{x^2}{x^2+y^2}\right|\le |x|.$$