Exercise 4.4.2

Question

\begin{enumerate}[label=(\alph*)] 
  \item Is $f(x)=1 / x$ uniformly continuous on $(0,1)$ ?
  \item Is $g(x)=\sqrt{x^{2}+1}$ uniformly continuous on $(0,1)$ ?
  \item Is $h(x)=x \sin (1 / x)$ uniformly continuous on $(0,1)$ ?
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item No, intuitively because slope becomes unbounded as we approach zero. Rigorously consider $x_n = 2/n$ and $y_n = 1/n$ we have $|x_n - y_n| 	o 0$ but $|1/x_n - 1/y_n| = |n/2 - n| = n/2$ is unbounded meaning $f$ cannot be uniformly continuous by Theorem 4.4.5.
  \item Yes, since it's continuous on $[0, 1]$ Theorem 4.4.7 implies it is uniformly continuous on $[0, 1]$ and hence on any subset as well.
  \item Yes, since $h$ is continuous over $[0,1]$ implying it is also uniformly continuous over $[0,1]$ by Theorem 4.4.7
   \end{enumerate}

Exercise 4.4.2

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Exercise 4.4.2

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