Can you use level sets to suggest discontinuity?

Question

Consider the function

$$f(x, y) = \frac{x^2 + y^2}{y}$$

for which I already showed that the level set of height $c$ is given by a circle of the form

$$C_c: (x - 0)^2 + (y - \frac{c}{2})^2 = \frac{c^2}{4}$$

Next, observe a plot of the level sets $C_1$ (red), $C_2$ (blue) and $C_3$ (green).

Note that the point $(0, 0)$ is contained in all three sets; in other words, $f$ seems to have more than one value at $(0, 0)$ (this, I haven't quite grasped yet).

Is it valid to make the argument that $f$ is discontinuous in $(0, 0)$, since it can be shown that some level sets intersect?

More generally, is any intersection between two level sets a point of discontinuity?

Answer 1

More generally, is any intersection between two level sets a point of discontinuity?

Yes. Of course we are talking about an intersection in the limiting sense.

Let $C_1$ denote the level set where $f(x,y) = c_1$, and $C_2$ the level set where $f(x,y) = c_2$ ($c_1 \neq c_2$). Suppose that $(0,0)$ is a limit point of $C_1$ and $C_2$ (i.e. the level sets 'intersect'). Then I can take the limits $$\lim_{(x,y) \to (0,0), (x,y) \in C_1}f(x,y) = c_1 \neq c_2 = \lim_{(x,y) \to (0,0), (x,y) \in C_2}f(x,y).$$

So $f$ cannot be continuous at $(0,0)$.