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A function  $  f :  E \rightarrow Y $
given by an equation  $  F ( x , y ) = z _ {0} $, 
where  $  F:  X \times Y \rightarrow Z $, 
$  x \in X $, 
$  y \in Y $, 
$  E \subset  X $, 
and  $  X $, 
$  Y $
and  $  Z $
are certain sets, i.e. a function  $  f $
such that  $  F ( x , f ( x) ) = z _ {0} $
for any  $  x \in E $. 
If  $  X $, 
$  Y $
and  $  Z $
are topological spaces and if  $  F ( x _ {0} , y _ {0} ) = z _ {0} $
for some point  $  ( x _ {0} , y _ {0} ) \in X \times Y $, 
then under certain conditions the equation  $  F ( x , y ) = z _ {0} $
is uniquely solvable in one of the variables in some neighbourhood of  $  ( x _ {0} , y _ {0} ) $. 
Properties of the solution of this equation are described by implicit-function theorems.

The simplest implicit-function theorem is as follows. Suppose that  $  X $
and  $  Y $
are subsets of the real line  $  \mathbf R $, 
let  $  x _ {0} \in X $, 
$  y _ {0} \in Y $, 
and let  $  ( x _ {0} , y _ {0} ) $
be an interior point of the plane set  $  X \times Y $; 
if  $  F $
is continuous in some neighbourhood of  $  ( x _ {0} , y _ {0} ) $, 
if  $  F ( x _ {0} , y _ {0} ) = 0 $
and if there are a  $  \delta > 0 $
and an  $  \epsilon > 0 $
such that  $  F ( x , y ) $, 
for any fixed  $  x \in ( x _ {0} - \delta , x _ {0} + \delta ) $, 
is strictly monotone on  $  ( y _ {0} - \epsilon , y _ {0} + \epsilon ) $
as a function of  $  y $, 
then there is a  $  \delta _ {0} > 0 $
such that there is a unique function

$$ 
f :  ( x _ {0} - \delta _ {0} , x _ {0} + \delta _ {0} )  \rightarrow \ 
( y _ {0} - \epsilon , y _ {0} + \epsilon )
$$

for which  $  F ( x , f ( x) ) = 0 $
for all  $  x \in ( x _ {0} - \delta _ {0} , x _ {0} + \delta _ {0} ) $; 
moreover,  $  f $
is continuous and  $  f ( x _ {0} ) = y _ {0} $.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050310a.gif" />

Figure: i050310a

The hypotheses of this theorem are satisfied if  $  F $
is continuous in a neighbourhood of  $  ( x _ {0} , y _ {0} ) $, 
if the partial derivative  $  F _ {y} $
exists and is continuous at  $  ( x _ {0} , y _ {0} ) $, 
if  $  F ( x _ {0} , y _ {0} ) = 0 $, 
and if  $  F _ {y} ( x _ {0} , y _ {0} ) \neq 0 $. 
If in addition the partial derivative  $  F _ {x} $
exists and is continuous at  $  ( x _ {0} , y _ {0} ) $, 
then the implicit function  $  f $
is differentiable at  $  x _ {0} $, 
and

$$ 

\frac{d f ( x _ {0} ) }{dx}
  =  -

\frac{F _ {x} ( x _ {0} , y _ {0} ) }{F _ {y} ( x _ {0} , y _ {0} ) }
 .
$$

This theorem has been generalized to the case of a system of equations, that is, when  $  F $
is a vector function. Let  $  \mathbf R  ^ {n} $
and  $  \mathbf R  ^ {m} $
be  $  n $- 
and  $  m $-
dimensional Euclidean spaces with fixed coordinate systems and points  $  x = ( x _ {1} \dots x _ {n} ) $
and  $  y = ( y _ {1} \dots y _ {m} ) $, 
respectively. Suppose that  $  F $
maps a certain neighbourhood  $  W $
of  $  ( x _ {0} , y _ {0} ) \in \mathbf R  ^ {n} \times \mathbf R  ^ {m} $(
$  x _ {0} \in \mathbf R  ^ {n} $, 
$  y _ {0} \in \mathbf R  ^ {m} $) 
into  $  \mathbf R  ^ {m} $
and that  $  F _ {i} $, 
$  i = 1 \dots m $, 
are the coordinate functions (of the  $  n + m $
variables  $  x _ {1} \dots x _ {n} , y _ {1} \dots y _ {m} $) 
of  $  F $, 
that is,  $  F = ( F _ {1} \dots F _ {m} ) $. 
If  $  F $
is differentiable on  $  W $, 
if  $  F( x _ {0} , y _ {0} ) = 0 $
and if the Jacobian

$$ 
\left .

\frac{\partial  ( F _ {1} \dots F _ {m} ) }{\partial  ( y _ {1} \dots y _ {m} ) }

\right | _ {( x _ {0}  , y _ {0} ) } \ 
\neq  0 ,
$$

then there are neighbourhoods  $  U $
and  $  V $
of  $  x _ {0} \in \mathbf R  ^ {n} $
and  $  y _ {0} \in \mathbf R  ^ {m} $, 
respectively,  $  U \times V \subset  W $, 
and a unique mapping  $  f :  U \rightarrow V $
such that  $  F ( x , f ( x) ) = 0 \in \mathbf R  ^ {m} $
for all  $  x \in U $. 
Here  $  f ( x _ {0} ) = y _ {0} $, 
$  f $
is differentiable on  $  U $, 
and if  $  f = ( f _ {1} \dots f _ {m} ) $, 
then the explicit expression for the partial derivatives  $  \partial  f _ {j} / \partial  x _ {i} $, 
$  i = 1 \dots n $, 
$  j = 1 \dots m $, 
can be found from the system of  $  m $
linear equations in these derivatives:

$$ 

\frac{\partial  F _ {k} }{\partial  x _ {i} }
 +
\sum_{j=}^ { m } 

\frac{\partial  F _ {k} }{\partial  y _ {j} }

\frac{\partial  f _ {j} }{\partial  x _ {i} }
  =  0 ,
$$

$  k = 1 \dots m $, 
$  i $
is fixed  $  ( i = 1 \dots n ) $. 
Sometimes the main assertion of the theorem is stated as follows: There are neighbourhoods  $  U $
of  $  x _ {0} $
in  $  \mathbf R  ^ {n} $
and  $  W _ {0} $
of  $  ( x _ {0} , y _ {0} ) $
in  $  \mathbf R  ^ {n} \times \mathbf R  ^ {m} $, 
$  W _ {0} \subset  W $, 
and a unique mapping  $  f :  U \rightarrow \mathbf R  ^ {m} $
such that  $  ( x , f ( x) ) \in W _ {0} $
and  $  F ( x , f ( x) ) = 0 $
for all  $  x \in U $. 
In other words, the conditions

$$ 
( x , y )  \in  W _ {0} ,\ \ 
F ( x , y )  =  0
$$

are equivalent to  $  x \in U $, 
$  y = f ( x) $. 
In this case one says that the equation  $  F ( x , y ) = 0 $
is uniquely solvable in the neighbourhood  $  W _ {0} $
of  $  ( x _ {0} , y _ {0} ) $.

The classical implicit-function theorem thus stated generalizes to the case of more general spaces in the following manner. Let  $  X $
be a topological space, let  $  Y $
and  $  Z $
be affine normed spaces over the field of real or complex numbers, that is, affine spaces over the relevant field to which are associated normed vector spaces  $  \mathbf Y $
and  $  \mathbf Z $, 
$  \mathbf Y $
being complete, let  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $
be the set of continuous linear mappings from  $  \mathbf Y $
into  $  \mathbf Z $, 
and let  $  W $
be an open set in the product space  $  X \times Y $, 
$  ( x _ {0} , y _ {0} ) \in W $, 
$  x _ {0} \in X $, 
$  y _ {0} \in Y $.

Let  $  F :  W \rightarrow Z $
be a continuous mapping and  $  F ( x _ {0} , y _ {0} ) = z _ {0} $. 
If for every fixed  $  x $
and  $  ( x , y ) \in W $
the mapping  $  F $
has a partial [[Fréchet derivative|Fréchet derivative]]  $  F _ {y} \in {\mathcal L} ( \mathbf Y , \mathbf Z ) $, 
if  $  F _ {y} ( x , y ) :  W \rightarrow {\mathcal L} ( \mathbf Y , \mathbf Z ) $
is a continuous mapping and if the linear mapping  $  F _ {y} ( x _ {0} , y _ {0} ) :  \mathbf Y \rightarrow \mathbf Z $
has a continuous inverse linear mapping (that is, it is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $), 
then there exist open sets  $  U \subset  X $
and  $  V \subset  Y $, 
$  x _ {0} \in U $, 
$  y _ {0} \in V $, 
such that for any  $  x \in U $
there is a unique element  $  y \in V $, 
denoted by  $  y = f ( x) $, 
satisfying the equations

$$ 
f ( x)  \in  V \ \ 
\textrm{ and } \  F ( x , f ( x) )  =  z _ {0} .
$$

The function  $  y = f ( x) $
thus defined is a continuous mapping from  $  U $
into  $  V $, 
and  $  y _ {0} = f ( x _ {0} ) $.

If  $  X $
is also an affine normed space, then under certain conditions the implicit function  $  f :  x \mapsto y $
which satisfies the equation

$$ \tag{1 }
F ( x , y )  =  z _ {0}  $$

is also differentiable. Namely, let  $  X $, 
$  Y $
and  $  Z $
be affine normed spaces, let  $  W $
be an open set in  $  X \times Y $, 
let  $  F :  W \rightarrow Z $, 
$  F ( x _ {0} , y _ {0} ) = z _ {0} $, 
$  x _ {0} \in Y $, 
and let  $  f $
be the implicit mapping given by (1), taking a certain neighbourhood  $  U $
of  $  x _ {0} $
into an open subset  $  V $
of  $  Y $, 
$  U \times V \subset  W $. 
Thus, for all  $  x \in U $,

$$ \tag{2 }
f ( x)  \in  V ,\ \ 
F ( x , f ( x) )  =  z _ {0} .
$$

Suppose also that  $  f $
is continuous at  $  x _ {0} $
and that  $  f ( x _ {0} ) = y _ {0} $. 
If  $  F $
is differentiable at  $  ( x _ {0} , y _ {0} ) $, 
if its partial Fréchet derivatives  $  F _ {x} ( x _ {0} , y _ {0} ) $
and  $  F _ {y} ( x _ {0} , y _ {0} ) $
are continuous linear operators taking the vector spaces  $  \mathbf X $
and  $  \mathbf Y $
associated with  $  X $
and  $  Y $
into the vector space  $  \mathbf Z $
associated with  $  Z $, 
and if the operator  $  F _ {y} ( x _ {0} , y _ {0} ) $
is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $, 
then  $  f $
is differentiable at  $  x _ {0} $
and its Fréchet derivative is given by

$$ 
f ^ { \prime } ( x _ {0} )  = \ 
- F _ {y} ^ { - 1 }
( x _ {0} , y _ {0} )
\circ F _ {x} ( x _ {0} , y _ {0} ) .
$$

This is obtained as a result of formally differentiating (2):

$$ 
F _ {x} ( x _ {0} , y _ {0} ) +
F _ {y} ( x _ {0} , y _ {0} )
\circ f ^ { \prime } ( x _ {0} )  = \ 
0  \in  {\mathcal L} ( \mathbf X , \mathbf Y )
$$

and multiplying this equality on the left by  $  F _ {y} ^ { - 1 } ( x _ {0} , y _ {0} ) $.

If in addition the mapping  $  F :  W \rightarrow Z $
is continuously differentiable on  $  W $, 
if the implicit function  $  f :  U \rightarrow V $
is continuous on  $  U $, 
$  U \times X \subset  W $, 
and if for any  $  x \in U $
the partial Fréchet derivative  $  F _ {y} ( x , f ( x) ) $
is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $, 
then  $  f $
is a continuously-differentiable mapping of  $  U $
into  $  V $.

In the general case one can also indicate conditions for the existence and the uniqueness of the implicit function in terms of the continuity of the Fréchet derivative: If  $  Z $
is complete, if the mapping  $  F :  W \rightarrow Z $
is continuously differentiable on  $  W $, 
if  $  F ( x _ {0} , y _ {0} ) = z _ {0} $, 
and if the partial Fréchet derivative  $  F _ {y} ( x _ {0} , y _ {0} ) $
is an invertible element of  $  {\mathcal L} ( \mathbf Y , \mathbf Z ) $, 
then (1) is uniquely solvable in a sufficiently small neighbourhood of  $  ( x _ {0} , y _ {0} ) $, 
i.e. there exist neighbourhoods  $  U $
of  $  x _ {0} $
in  $  X $
and  $  V $
of  $  y _ {0} $
in  $  Y $, 
$  U \times V \subset  W $, 
and a unique implicit function  $  f :  U \rightarrow V $
satisfying (2). Here  $  f $
is also continuously differentiable on  $  U $. 
In this form the implicit-function theorem for normed spaces is a direct generalization of the corresponding classic implicit-function theorem for a single scalar equation in two variables.

Furthermore, if  $  F :  W \rightarrow Z $
is a  $  k $-
times continuously-differentiable mapping in a neighbourhood  $  W $
of  $  ( x _ {0} , y _ {0} ) $, 
$  k = 1 , 2 \dots $
then the implicit function  $  f :  U \rightarrow V $
is also  $  k $
times continuously differentiable.

Far-reaching generalizations of the classic implicit-function theorem to differential operators were given by J. Nash (see [[Nash theorems (in differential geometry)|Nash theorems (in differential geometry)]]).

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====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elemente der Funktionalanalysis" , Akademie Verlag  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Schwartz,   "Cours d'analyse" , '''1''' , Hermann  (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1''' , MIR  (1982)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Fleming,   "Functions of several variables" , Addison-Wesley  (1965)</TD><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Flett,   "Differential Analysis" ,  Cambridge University Press  (1980)</TD></TR></table>