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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.