Hello im studying calculus at the university and I dont know how to solve the following exercise:
Study the continuity of the next function:
$$f(x,y) = \begin{cases} \frac{x^2-xy}{x+y}&\text{for } x+y\ne0\\ 0 &\text{for }(x,y) =(0,0). \end{cases}$$
I've tried to resolve it with iterated limits and directional limits, but im sure if its correct.
Best Answer
Note that $f(0,t) = 0$ (for $t \neq 0$), but $$f(t,-t+t^2) = \frac{t^2+t^2-t^3}{t^2} = 2-t,$$ again for $t \neq 0$. What happens when $t \to 0$? This shows that $f$ is not continuous at $(0,0)$.
On the other hand, $f$ is continuous everywhere else where $f$ is defined, since the numerator and denominator clearly are continuous.