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Problem:


Let \(f\) and \(g\) be two functions from \([0,1]\) to \([0,1]\) with \(f\) strictly increasing. Which of the following statements is always correct?

A. If \(g\) is continuous, then \(fog\) is continuous

B. If \(f\) is continuous, then \(fog\) is continuous

C. If \(f\) and \(fog\) is continuous, then \(g\) is continuous

D. If \(g\) and \(fog\) are continuous, then \(f\) is continuous


Discussion:


A: Let \(g(x)=x\) for all \(x\in [0,1]\).

\(f(x)=x\) for \(x\in [0,\frac{1}{2}]\) and \(f(x)=5+x\) for \(x\in (\frac{1}{2},1]\).

Then \(fog=f\) and \(f\) is not continuous.

So A is False.

B: Reverse \(f\) and \(g\) in A to show that B is False.

C: If \(f\) and \(fog\) are continuous then \(f\) is 1-1 (increasing), continuous map \([0,1]\to [0,1]\).

\(A \subset [0,1] \) be closed. Then \(A\) is compact. (Closed subsets of compact spaces are compact).

Therefore \(f(A)\) is compact. (continuous image of compact set is compact).

We have that \(f(A)\) is a compact subset of \([0,1]\). Therefore \(f(A)\) is closed in \([0,1]\). (compact subspace of Hausdorff space is closed).

Therefore, \(f\) is a closed map. So \(f^{-1}\) is continuous.

Hence \(f^{-1}ofog=g\) is continuous.

So, C is True.

D: Let \(g(x)=\frac{x}{4}\) for all \(x\in [0,1]\).

\(f(x)=x\) for \(x\in [0,\frac{1}{2}]\) and \(f(x)=5+x\) for \(x\in (\frac{1}{2},1]\).

Then \(fog(x)=f(\frac{x}{4})=\frac{x}{4}\) for all \(x\in [0,1]\).

So \(fog\) is continuous but \(f\) is not continuous.

So, D is False.


Chatuspathi

  • What is this topic: Real Analysis
  • What are some of the associated concept: Continuity,Closed Set, Compact Set
  • Book Suggestions: Introduction to Real Analysis by R.G. Bartle, D.R. Sherbert