Though more general, the theorem is actually a consequence of the inverse function theorem. If the function satisfies the equation $f(a) = b$, then the inverse of this function satisfies $g(b) = a$.
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{\displaystyle \square }
Similarly, there is the implicit function theorem for holomorphic functions.
The Fourier transform may be defined on the space of tempered distributions
S
(
R
n
)
{\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}
by duality of the Fourier transform on the space of Schwartz functions.
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Suppose also that the above equation is very difficult to solve (numerically) for a given y0, but easy to solve for a value y~ ”near” y0. Thus
(
x
n
)
{\displaystyle (x_{n})}
is a Cauchy sequence tending to
x
{\displaystyle x}
. Let us take the square matrix AWhere a, b, c, and d represents the number. \(y=\frac{x+1}{x}\)Upon swapping the variables,\(x=\frac{y+1}{y}\)xy = y + 1xy y = 1y(x-1) = 1\(y=\frac{1}{x-1}\)\(f^{-1}(x)=\frac{1}{x-1}\)Example 2. Then there exists an open neighbourhood V of
F
(
0
)
{\displaystyle F(0)\!}
in Y and a continuously differentiable map
G
:
V
X
{\displaystyle G:V\to X\!}
such that
F
(
G
(
y
)
)
=
y
{\displaystyle F(G(y))=y}
for all y in V.
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.