Describe the graph of the function: Level Sets, Intersections Describe the graph of the function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow |y|$$ computing some level sets and some intersections.
I have done the following:
The level curves are defined by $$\{(x, y) \mid |y|=c\}$$

*

*For $c=0$ we have that $|y|=0 \Rightarrow y=0$. So, for $c=0$, the level set consists of the $x-$axis.


*For $c<0$, the level set is the empty set.
Is this correct??
Could I improve something??

*

*For $c>0$ we have that $y=c, \text{ if } y>0$ and $y=-c, \text{ if } y<0$. So, of what consists the level set for $c>0$??

How can we compute the intersections??
EDIT:
In my book there is the following definition:
The intersection of the graph of $f$ is the intersection of the graph with a vertical plane.
For example, if we have $f(x, y)=x^2+y^2$ we have the following:
If $P_1$ is the plane $xz$ in $\mathbb{R}^3$ that is defined by $y=0$, then the intersection of $f$ is the set $$P_1 \cap \text{ graph } f=\{(x, y, z) \mid y=0, z=x^2\}$$
that is a parabola in the plane $xz$.
Similarily, if $P_2$ is the plane $yz$, that is defined by $x=0$, then the intersection $$P_2 \cap \text{ graph } f=\{(x, y, z) \mid x=0, z=y^2\}$$ is a parabola in the plane $yz$.
So, do we take which vertical plane we want??

As you wrote, the level set for a constant $c$ is the set of points $(x, y)$ such that $f(x, y) = c$. Your characterizations for $c = 0$ and $c < 0$ are totally correct.
For $c > 0$, again as you wrote, the level set is the set of all points such that $y = c$ or $y = -c$. So the level set will be the union of two parallel lines: the lines $y = c$ and the line $y = -c$.
