Let f : [a, b] → [a, b] such that f(f(x)) = and f(x) # x for all x e [a, b]. Show that ƒ isdiscontinuous at infinitely many points. (Hint: try proving it by contradiction)

Given that f:a,b→a,b such that f(f(x))=x and f(x)≠x for all x. The objective is to show that f is…

Answered: Let f : [a,b] discontinuous at…

Let f : [a,b] discontinuous at infinitely many points. (Hint: try proving it by contradiction) + [a, b] such that f(f(x)) = x and f(x) # x for all x e [a, b]. Show that f is

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 24E

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Let f : [a, b] → [a, b] such that f(f(x)) = and f(x) # x for all x e [a, b]. Show that ƒ is
discontinuous at infinitely many points. (Hint: try proving it by contradiction)

Expert Solution

Step 1

Given that $f : [a, b] \to [a, b]$ such that $f (f (x)) = x$ and $f (x) \neq x$ for all x.

The objective is to show that f is discontinuous at infinitely many points.

Assume that $f : [a, b] \to [a, b]$ such that $f (f (x)) = x$ and $f (x) \neq x$ for all x then f is continuous at infinitely many points.

Since f is continuous,

$\begin{array}{rcl} f (\underset{x \to a}{l i m} f (x)) & = & f (f (a)) \\ = & a \end{array}$

Take $y = f (x)$

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