A Inverse Function Theorem

Let f be a continuously differentiable function of x, both having the same dimension n greater than zero. If the derivative f (x) is non-zero at x = xq for which yo = f(xo), then there exists a continuous inverse function f (y), which maps an open set Y containing yo to an open set X containing xq. 

Note that an open set has each member completely surrounded by members of the same set (see Section 9.3, p. 268). Moreover, the inverse function is differentiable, the proof of which can be found in Rudin (1976). 

Under the given conditions, the theorem assures that y = f(x), which is a set of n equations 

Based on the given function f, we will develop an auxiliary function g. We will show it to be a contraction, which is associated to a unique fixed point. This property will then lead to the existence of the inverse function f . We start with the description of a contraction and its fixed point. 

A contraction is defined to be a function (f that maps a region X to itself 

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