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

[Eclair_de_XII]

TL;DR

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 $\mathbb{R}^2$.

 

[Office_Shredder]

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.

 

[FactChecker]

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.