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A contraction mapping is a type of function used in mathematics that shrinks distances between points in a metric space. It is utilized in proving Fixed Point Theorem, which states that every contraction mapping on a complete metric space has a unique fixed point. However, not all mapping functions are contraction maps.

An example of a non-contraction mapping is the function f(x) = x² on the interval [0,1]. To prove that this is not a contraction map, we can apply the definition of a contraction map. A function f(x) is a contraction map if there exists a constant k, 0<k<1, such that for any two points x and y in the interval [0,1], the distance between f(x) and f(y) is less than k times the distance between x and y.

Let`s take two points x=0 and y=1/2 in the interval [0,1]. The distance between them is 1/2. Now, applying the function f, we get f(x) = 0 and f(y) = 1/4. The distance between them is 1/4, which is not less than k times the distance between x and y for any k<1. Therefore, f(x) = x² is not a contraction map on the interval [0,1].

Another example of a non-contraction mapping is the function f(x) = sin(x) on the interval [0,π/2]. This function is not a contraction map because it is not a decreasing function on the interval. In order to be a contraction map, the function must be strictly decreasing. Therefore, f(x) = sin(x) is not a contraction map on the interval [0,π/2].

In conclusion, not all mapping functions are contraction maps. The function f(x) = x² and the function f(x) = sin(x) are examples of non-contraction mappings. These examples show that it is important to apply the definition of contraction mapping properly to determine whether a given function is a contraction map or not.