Given a function $f(x,y)$, can two different level curves of $f$ intersect? Why or why not?

Question

This is the question, and I think the answer is "yes." But the solution says "no." I don't know why it can't.

If $f(x,y)=y/x$, their lever curves intersect at $(0,0)$, don't they?

Answer 1

Two level curves can, by definition, not intersect. One level curve is defined as $f(x,y)=c_1$, the other as $f(x,y)=c_2$. If $c_1\neq c_2$ (else they are the same curve), if there exists a point on both level curves, that would mean $f(x,y)=c_1$ and $f(x,y)=c_2$, meaning $c_1=f(x,y)=c_2$ which cannot be true.

It is possible, however, to have one level curve which is composed of more than one 'line'. For example, for $f(x,y)=xy$, the level line for $f(x,y)=0$ is composed of both the line $x=0$ and $y=0$. This does not mean, however, that this line intersects any line.

Answer 2

Imagine yourself walking on the surface of your function. Travel to the point of intersection of the level curves and ask yourself "How high am I"?.

Answer 3

Several different level curves can "meet" at a common point (but only at points where $f$ fails to be continuous) --- as in your example. Strictly speaking though, those common points only lie on the closure of the level curves, not on the level curves themselves.

Answer 4

As mentioned, if the level curves intersect, then the two constant values involved must be equal to the value of $f$ in the intersection, and therefore with each other, so that these curves are not actually two "level curves" but one "level set". The definition may be the same, but these sets can be different than a regular curve.

In your example $f(x,y)=y/x$, the level curves do not contain the origin.

Another example: if $f(x,y)=x^4-x^2+y^2$, then the level curves for $0.1$ are two separate closed curves. See Wolfram Alpha plot