Homework Soursea let f be a continuous function whose domain is the closed
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]
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]
Solution
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<=c<=1 0<=f(c)<=1 f(0)<=f(c)<=f(1)![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](/WebImages/39/a-let-f-be-a-continuous-function-whose-domain-is-the-closed-1120065-1761595863-0.webp "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")
