Can a series of level sets define a function? Question

For each real number $c\in \mathbb{R}_+$ there is a corresponding curve of dimension $\mathbb{R}^{n-2}$ contained in the corresponding hyperplane $x_n=c$ (aka a plane curve). Does there exist a function $f$ such that each plane curve is a level curve of $f$ and $f$ is continuous.

Rephrased to be more concise, If  $\forall c \in \mathbb{R}_+ \exists f_c(x_1,...,x_{n-1})=c $ there exists $f:\mathbb{R}^{n-1}_+ \to \mathbb{R}_+$ whose level curves are given by the functions $f_c$ and $f\in C^0$
From intuition there should be an $f$ such that there are appropriate level curves. The much stricter condition should be that $f$ is continuous. I believe the best way to ensure that $f$ is continuous is to derive a necessary condition using the neighbourhood definition of continuity (If there is a better definition of continuous that works better please tell me).
Based on the neighbourhood definition of continuity, if for any neighbourhood $N_1(f_c(x))$ (where $x\in \mathbb{R}^n$) there exists a neighbourhood $N_2$ such that $f_{c2}(x)\in N_1(f_c(x))$ (where $c_2$ and $c_1$ are different values of $c$) whenever $x\in N_2$.
Notes


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*If you have any questions or need clarification please ask.

*If there is any inconsistency between my original question and my rephrasing please tell me.

 A: I do not think your reprhasing works, but I also think the original problem statement is rather strange. It seems to be using more dimensions than it needs to (or even really can).
Example A: Consider the example of the family of curves $\{\gamma_c\}$ in $\mathbb R^2$ in which for each $c \in\mathbb R_+,$ the curve $\gamma_c$ comprises the points $(x,y)\in\mathbb R^2$ that satisfy $x^2 + y^2 = c.$
That is, $\gamma_c$ is a circle of radius $\sqrt c$ centered at the origin.
Then each $\gamma_c$ is a level curve of the function $f(x,y) = x^2 + y^2.$
Now here's where the problem statement gets weird. The curves in Example A all have dimension $1.$ The problem statement says the curves have dimension $n-2.$ (Actually it says "dimension $\mathbb R^{n-2}$," which seems a strange way to specify the number of dimensions of a curve. Curves of dimension $n-2$ are usually embedded in spaces of higher dimension than $\mathbb R^{n-2}$. But I digress.)
The problem statement then mentions the hyperplane $x_n = c,$ which says that the hyperplanes in Example A are the planes $x_3 = c.$ Presumably (though really, the problem statement should have been more explicit about this), we are supposed to assume that $x_3$ is the third coordinate of a Cartesian space, at least $\mathbb R^3$ if not something higher. Yet Example A--function, level curves, and all--lives entirely in $\mathbb R^2.$ Where does a third dimension come from?
I suppose the way we're supposed to interpret Example A for the purpose of the problem statement is that if we plot the graph $z = f(x,y)$ in $\mathbb R^3,$
where $f(x,y) = x^2 + y^2,$ we get a paraboloid whose slices in each of the hyperplanes $x_3 = c$ is a circle of radius $\sqrt c$ around the point $(0,0,c).$
My objection to this is that a painting of a pipe is not a pipe, and a graph of a function is not a function. The level curves of a function over $\mathbb R^2$ all lie in $\mathbb R^2.$ They are not scattered over multiple hyperplanes of $\mathbb R^3.$
Of course, you could map such a set of curves in $\mathbb R^3$ back to $\mathbb R^2,$ but why introduce such a complication in the first place?
Another weakness in the problem statement is that it uses vague language. It says there is a mapping from $\mathbb R_+$ to the set of curves of some dimension. Fine, we have two sets, there exists a mapping between them, that's clear.
Next it asks if there is a continuous function $f$ such that these curves are level curves of $f.$
Wait, which curves? The curves from some mapping the problem-setter has in mind, which we're supposed to guess? Are we supposed to say whether there is such an $f$ for every mapping from $\mathbb R_+$ to curves of dimension $n-2$?
Or just whether there exists some such mapping? 
A problem statement that would make more sense to me would be:

Given that for each $c \in \mathbb R_+$  there is a corresponding curve of dimension $n-2$ in $\mathbb R^{n-1},$ does there necessarily exist a continuous function $f$ such that each of those curves is a level curve of $f$?


Now to try to give some hints for answering the question rather than picking it apart:
The problem statement says very little about the properties of the curves corresponding to each $c$ in $\mathbb R_+.$
All it says is that the curve corresponding to each $c$ has dimension $n-2$ and lies in the hyperplane $x = c.$
The level curves of a single continuous function $f$ have properties (by themselves, and relative to each other) that they must satisfy. Where in the problem statement does it say that the curves that it describes have those properties?
Supposing that the point of the problem is to determine whether the mapping to curves in the first part necessarily gives a set of level curves of a continuous function,
you can either prove that the properties given in the first part of the problem imply all the necessary properties of a set of level curves, in which case the answer is "yes,"
or you can find a mapping to curves such that the curves satisfy the requirements given in the first part of the problem but are not a set of level curves of a continuous function, in which case you have a counterexample and the answer is "no."
One last hint: this might go easier if you try first to find a counterexample.
