Editing Function of a real variable (section)

==Implicit functions==

A '''real-valued [[implicit function]] of a real variable''' is not written in the form "''y'' = ''f''(''x'')". Instead, the mapping is from the space <math>\mathbb{R}</math><sup>2</sup> to the [[zero element]] in <math>\mathbb{R}</math> (just the ordinary zero 0):

:<math>\phi: \R^2 \to \{0\} </math>

and

:<math>\phi(x,y) = 0 </math>

is an equation in the variables. Implicit functions are a more general way to represent functions, since if:

:<math>y=f(x) </math>

then we can always define:

:<math> \phi(x, y) = y - f(x) = 0 </math>

but the converse is not always possible, i.e. not all implicit functions have the form of this equation.