composition of two uniformly continuous functions.

Question

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ are two uniform continuous functions. Which of the following options are correct and why?

1. $f(g(x))$ is uniformly continuous.

2. $f(g(x))$ is continuous but not uniformly continuous.

3. $f(g(x))$ is continuous and bounded.

My attempt:

Every uniformly continuous function maps a Cauchy sequence to a Cauchy sequence. (Here I have a doubt, as the converse may not be true). So if $\{x_n\}$ be a Cauchy sequence, $\{f(x_n)\}$ and $\{g(f(x_n))\}$ both will be Cauchy sequence. So $g(f(x))$ will be uniformly continuous, i.e. 1 is true.

Composite function of two continuous functions will be continuous. As 1 is true, 2 is false.

$f(x) = x $ is uniformly continuous. $g(x) = \log(x)$ is uniformly continuous in $[1,\infty)$. So $g(f(x)) = \log(x)$ is uniformly continuous in $[1, \infty)$, but not in $\mathbb{R}$.

Thank you for your help.

Answer 1

• Option $(1):$ True:

  Choose $\epsilon>0.$

  Then $\exists~\delta_1>0$ such that $|x_1-x_2|<\delta_1$$\implies|f(x_1)-f(x_2)|<\epsilon.$

  For above $\delta_1>0~\exists~\delta>0$ such that $|x_1-x_2|<\delta$$\implies|g(x_1)-g(x_2)|<\delta_1.$

  Consequently, $|x_1-x_2|<\delta$$\implies|(f\circ g)(x_1)-(f\circ g)(x_2)|<\epsilon.$

• Options $(2),~(3):$ Not necessarily true:

  Take $f(x)=g(x)=x$ on $\mathbb R.$

  Then $f,g$ are uniformly continuous on $\mathbb R.$

  Note $(f\circ g)(x)=x$ is unbounded and uniformly continuous on $\mathbb R.$

Answer 2

Your suspicion is justified. The argument you give for the validity of 1 is faulty since it only shows the composite preserves Cauchy sequences, but it does not show that the composite is uniformly continuous. Try working directly with the definition of uniform continuity instead to give a direct proof.

You can make things easier with the counter example to 3: Take $f(x)=g(x)=x$ on $\mathbb R$.