Suppose that f is a function that maps [a,b] onto [a,b] and f is continuous. Pro

Question

Suppose that f is a function that maps [a,b] onto [a,b] and f is continuous. Prove that there is an x[a,b] such that f(x)=x

Explanation / Answer

Conditions of Intermediate Value Theorem (IVT): 1. f(x) is continuous on the interval [a, b] 2. a > b Therefore, the function f(x) = x fits the IVT conditions and we can proceed to the proof using this theorem. Since f(x) maps [a,b] onto [a,b], we know that f(x) = x for all x. This means that our function is y = x Therefore f(a) = a and f(b) = b Thus, there must exist some value x on the interval [a, b] such that f(x) = x. ---- Plain words explanation: The function is f(x)=x, which is the same thing as y=x. This means that at any value x - and there are MANY values of x on the interval [a,b] - there are values that correspond to this function.

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