Why are function definitions not written with the $:=$ sign instead of the $=$ sign. It seems to me that $:=$ would be more intuitive and avoid a lot of unnecessary ambiguity.
Consider the following example:
$$f(x,y,z) = x^2 + y^2 + z^2$$ $$f(x,y,z) = 0$$
The first line - defines a real valued function who's output is the sum of squares of all of its parameters $x,y,z$
The second line - either also defines a real valued function who's output is always $0$, or more commonly, is asking us to find the roots of some function $f(x,y,z)$
Notice the subtle difference between the meaning of the 2 equal signs. So, would it not be more appropriate to instead write the first line (the actual function definition and not root finding) with the $:=$ sign? (since we are defining it, after all)
$$f(x,y,z) := x^2 + y^2 + z^2$$ $$f(x,y,z) = 0$$
Is there a reason why we don't do this? Am I totally mistaken on some subtle detail in the notation?