Editing Implicit function (section)

==Caveats==
Not every equation {{math|1=''R''(''x'', ''y'') = 0}} implies a graph of a single-valued function, the circle equation being one prominent example. Another example is an implicit function given by {{math|1=''x'' − ''C''(''y'') = 0}} where {{mvar|C}} is a [[cubic polynomial]] having a "hump" in its graph. Thus, for an implicit function to be a ''true'' (single-valued) function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a true function only after "zooming in" on some part of the {{mvar|x}}-axis and "cutting away" some unwanted function branches. Then an equation expressing {{mvar|y}} as an implicit function of the other variables can be written.

The defining equation {{math|1=''R''(''x'', ''y'') = 0}} can also have other pathologies. For example, the equation {{math|1=''x'' = 0}} does not imply a function {{math|''f''(''x'')}} giving solutions for {{mvar|y}} at all; it is a vertical line. In order to avoid a problem like this, various constraints are frequently imposed on the allowable sorts of equations or on the [[function domain|domain]]. The [[implicit function theorem]] provides a uniform way of handling these sorts of pathologies.