a) let f be a continuous function whose domain is the closed interval [0,1] and

Question

a) let f be a continuous function whose domain is the closed interval [0,1] and whose co-domain is also the closed interval [0,1]. Prove that there exists some c in [0,1] such that f(c) = c.

Give a geometrical interpretation of part a in terms of the graph of the curve y = f(x)

Give an example to show that the result of part a is not true if f is not continuous, that is, give an example of a function whose domain is the closed interval [0,1] and whose codomain is also [0,1] but f is not continuous and f(x) cannot equal x for any x in [0,1]

Explanation / Answer

Let f be a continuous function who's domain is the closed interval [0,1] and whose co-domain is also the closed interval [0,1]. Prove that there exists some value c in [0,1] such that f(c)=c . Hints Provided: 1) consider the function g(x)=x-f(x) . Argue that g is continuous on [0,1] . Draw some inference about g(0) and g(1), specifically, are they positive or negative. Apply Intermediate Value Theorem to the function g. What I have so far: F(x) must be equal to x, ie. f(x)=x 0

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