How do you show that the following multivariable function is continuous?
$f(x,y) = \frac{2x^2y}{x^4 + y^2}$
I think I want to show that for every point $(x_0,y_0)$:
$\forall \varepsilon >0, \ \exists\delta >0$ such that $||(x,y)-(x_0,y_0)|| < \delta \implies |f(x,y) - f(x_0,y_0)| < \varepsilon $
But I don't really get how to apply this.
That's indeed what you want to show but there is an easier way to prove continuity. What I would do is use the fact that composition, multiplications and additions of continuous functions are continuous. And since you know that $\frac{1}{x}, 2x^2 y$ and $x^4 + y^2$ all are continuous on $\mathbb{R}^2 \setminus \{(0,0) \}$ that should be enough