Implicit function theorem

In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x , y ) = 0 {\displaystyle F(x,y)=0} can also be specified as the graph of a function f {\displaystyle f} , so that for each point ( x , y ) {\displaystyle (x,y)} on part of the curve, one has y = f ( x ) {\displaystyle y=f(x)} . An example is the unit circle, whose points ( x , y ) {\displaystyle (x,y)} satisfy x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} , which can locally be solved (if y > 0 {\displaystyle y>0} ) by y = 1 − x 2 {\displaystyle y={\sqrt {1-x^{2}}}} , expressing the top semicircle as a graph.

Source: Wikipedia — Implicit function theorem (CC BY-SA 4.0)