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November 14, 2013

Fixed Point of continuous bounded function

Let's understand Fixed Point of continuous bounded function with the help of a problem. This problem is useful for College Mathematics.

f: [0 , \infty ) to [0. \infty ) is continuous and bounded then f has a fixed point.

True

Discussion: Consider the function g(x) = f(x) - x. Since f(x) and x are continuous then g(x) must be continuous. Since f(x) is bounded then there exists a M such that f(x) < M.

Now f(0) \ge 0 as the codomain is [0, \infty ) . Thus g(0) = f(0) - 0 \ge 0 . Also g(M) must be negative as f(M) < M. Since g(x) is continuous, by Intermediate Value Property of Continuous Functions g(x) must attain the value of 0 somewhere between x = 0 to x = M. Suppose that value is c.

Hence g(c) = f(c) - c = 0 or f(c) = c.

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