Is a plane $z=x+y$ a function?

Question

Can we think of plane in three-dimensional space as a function of 2 variables? In other words, is the plane $z=f(x,y)=x+y$ a function?

Answer 1

A plane is not a function.

No function is a plane.

Both are of different nature. A function is a relation that uniquely associates members of one set with members of another set (multivariate function in the present case). A plane is a geometrical object, a flat two-dimensional surface.

In a given system of coordinates, a plane can be mathematically described by a function which is loosely called "the equation of the plane" relatively to this system of coordinates.

In a given system of coordinates the "equation" of a plane is a function. In a different system of coordinates the "equation" of the same plane is a different function.