Increasing function and continuity (TIFR 2015 problem 7)

Question:

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.

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