Constrain a multivariate function.

Question

If given a function $F(x,y)$ of two separate and independent variables $x$ and $y$, would it be possible to define a relation between the two variables that is only true for certain combinations of values, without changing their independence.

For example,

I define that for certain pairs of values that

$x=y$

and define a function $F_r(x,y)$ on the domain A containing tuples $(a_0,a_0)$ for any real number $a_0$

As we've limited the domain of $F_r$ even though it should be a multivariate function, and it takes two arguments it's domain is limited so that it can't take any two choices of $x$ or $y$, can this be described as a multivariate function anymore?

We also see that $F_r$ is not really continuous in all directions on its domain $A$ and we find that we cannot define the partial derivatives, this is much like a function where we explicitly define a relation for all $x$ and $y$ and define $F(x,y)$ and it makes sense in this case, for some combination $x,y$ the relation

$F(x,y)=F_r(x,y)=g(x)=g(y)$

should hold in this case.

Is it acceptable for two independent variables to introduce a constraint like this which is true only for when they have values such that they follow this relation? Would I say that $x$ and $y$ are dependent (in that context) when we are only interested in the particular set of tuples $(x,y) ∈ A$ or that they simply act as dependent if we were to only consider relationships that are true in the case that the values of $x$ and $y$ fall under this given relation, but that they are independent in general?

Answer 1

The above Question is a continuation of a thread started by the OP in their previous Question. Here is the original discussion migrated over (to tidy up the duplication):

When dealing with a function $F(x,y)$, I wish to analyze when its arguments are equal.

Given a function $F(x,y)=xy,$ both $x,y$ are 'varying' in this case. $F(x,y)$ cannot be expressed as a single variable function $g$ (which takes one argument) application to any one variable (they are 'independent').

When we define $x=y$ for $F(x,y),$ we run into a slightly confusing statement, if for $x$ or $y$ we want to analyze them for differing values, then based on this relationship (in this context) they seem to be not independent anymore.

The domain of the given function $F$ is the plane $\mathbb R^2;$ as such, its inputs $x$ and $y$ are independent of each other.

When the dependency $x=y$ is imposed as a constraint, a new function $F_r$ (called the restriction of $F$) is created. The inputs of $F_r$ mutually depend on each other, so its domain is a proper subset of $\mathbb R^2.$

To be clear:

1. $F$ has domain $\mathbb R^2$ and rule $F(x,y)=xy$
2. $F_r$ has domain $\{(x,x)\vert x\in\mathbb R\}$ (a line in $\mathbb R^2$) and rule $F_r(x,y)=xy$
3. $g$ has domain $\mathbb R$ and rule $g(x)=x^2.$