# Is the level-set for a given function unique?

Summary:
Assume that I've only taken math courses up until Calculus III. Informal definition is as follows:

Let ##f:\mathbb{R}^2\longrightarrow\mathbb{R}## be defined. Then the level set of ##f## w.r.t. some point ##z\in f(\mathbb{R}^2)## is the set of points ##(x,y)## in the ##xy##-plane with the property that ##f(x,y)=z##.
Let ##f,g:\mathbb{R}^2\longrightarrow\mathbb{R}## be defined, and denote ##D=f(\mathbb{R}^2)##. Assume without loss of generality that ##g(\mathbb{R}^2)\equiv f(\mathbb{R}^2)##.

Define a function ##\varphi_f:D\longrightarrow \mathbb{R}^2## as follows: ##\varphi_f(z)=\{(x,y):f(x,y)=z\}##, and define ##\varphi_g## similarly. Let ##z\in D##. Suppose that ##\varphi_g\equiv\varphi_f##. Then ##\{(x,y):f(x,y)=z\}=\{(x,y):g(x,y)=z\}##. In other words, the point ##z## associates the functions ##f,g## to the same exact points ##(x,y)## in ##\mathbb{R}^2##.

Hence, for every ##(x,y)\in \varphi_f(z)##:

\begin{eqnarray*}
(x,y,f(x,y))=(x,y,g(x,y))\\
(0,0,(f-g)(x,y))=(0,0,0)
\end{eqnarray*}

It follows then, that ##f\equiv g##.

This isn't actually a proof; I'm just asking a question that I cannot find an answer to on the internet. This is just a rudimentary explanation of why I think level sets are unique in ##\mathbb{R}^2##.

Last edited:
• Delta2

## Answers and Replies

Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
I think this is right. If you are given all the level sets, along with the value f takes on each of them, f is clearly uniquely defined as given any point (x,y), it must be contained inside a single level set, and then f(x,y) is uniquely determined.

I have typically seen the word "level set" to refer to just the sets of (x,y) without z attached. For example ##f(x,y)=x^2+y^2## has circles as its level sets. Just being told the level sets are circles is *not* sufficient to uniquely identify the function.

• Delta2 and Eclair_de_XII
FactChecker
Science Advisor
Gold Member
This is not a proof that the level sets are unique. If anything is as an attempted proof that the level sets uniquely define a function.
Lesson #1 in math after Calc III: "Informal" definitions and proofs are dangerous. You have to be careful with your words.