I've been given the following function of two variables:
$$f(x, y) = \frac{xy \left(2x^2 - y^2 \right)}{x^2 + 2y^2}$$
and was asked to determine whether or not $f_x(x, y)$ and $f_y(x, y)$ are continuous at $(0, 0)$ or not. The answer states that both of the partials are continuous at $(0, 0)$, but not differentiable. How is this possible? If $f(0, 0)$ isn't even defined, how could the partial derivatives even exist at $(0, 0)$, let alone be continuous? I don't believe I'm fully understanding the notion of continuity here.
Edit: I'm starting to think that the problem must have omitted something; if you attempt to apply the definition of the partial derivative at $(0, 0)$:
$$f_x(0, 0) = \lim_{x \rightarrow 0} \frac{f(x, 0) - f(0, 0)}{x}$$
you will immediately notice that this limit cannot possibly exist, because $f(0, 0)$ is undefined, which confirms my original suspicions.