Find all $(x,y)$ for which $f(x,y)$ is continuous $$f(x) = \begin{cases} \dfrac{x^2+xy+y^2}{x^2+y^2} &\text{ if $(x,y)\neq (0,0)$}\\1 &\text{ if $(x,y)=(0,0)$}\end{cases}$$
When I evaluate, using polar coordinates, $$\lim_{(x,y)\to(0,0)}\dfrac{x^2+xy+y^2}{x^2+y^2}=\lim_{\rho \to0}1+\cos\theta\sin\theta$$ and therefore, the limit does not exist at $(0,0)$.
Knowing this, what would be my conclusion? What is the answer to the question? Would I say that it is continous at all points other than $(0,0)$?

yes exactly it’s continuous with exclusion of (0,0)
