CRN 11378: HW 5

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(Created page with "'''Problem 21.''' Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$. # Show that $f$ is continuous at $0$. # Show that $f$ is continuous at any $x_ne...")
 
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'''Problem 21.''' Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.
 
'''Problem 21.''' Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.
 
# Show that $f$ is continuous at $0$.
 
# Show that $f$ is continuous at $0$.
# Show that $f$ is continuous at any $x_neq0$. (The identity $a^3-b^3=)a-b)(a^2+ab+b^2)$ will be helpful.)
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# Show that $f$ is continuous at any $x_0\neq 0$. (The identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ will be helpful.)
  
'''Problem 17.''' Prove: A subset $F\subseteq \mathbb{R}$ is closed if and only if every Cauchy sequence contained in $F$ converges to an element in $F$.
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'''Problem 22.''' Assume $f:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$. Show that $\{x\in\mathbb{R}\ |\  f(x)=0\}$ is a closed set.
  
'''Problem 18.''' Find all accumulation points of the set
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'''Problem 23.''' Let $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.
\[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\]
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#Show that $f$ is uniformly continuous on $\mathbb{R}$.
Remember that $A=B\ \Leftrightarrow\  (A\subseteq B)\wedge (B\subseteq A)$.
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#Now assume that $0<c<1$. Show that there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y)))\ldots$.
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#Show that the result in 2. above may fail if $c=1$.
  
'''Problem 19.''' Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.
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'''Problem 24.''' A function $f:\mathbb{R} \to \mathbb{R}$ is called ''bounded'' if there is an $M>0$ such that $|f(x)|\leq M$ for all $x\in\mathbb{R}$.
 
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#Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\mathbb{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$.
'''Problem 20.''' Consider the following sets:
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#Show that the result may fail without the boundedness condition.
\[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\]
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For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.
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Revision as of 10:29, 5 November 2019

Problem 21. Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.

  1. Show that $f$ is continuous at $0$.
  2. Show that $f$ is continuous at any $x_0\neq 0$. (The identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ will be helpful.)

Problem 22. Assume $f:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$. Show that $\{x\in\mathbb{R}\ |\ f(x)=0\}$ is a closed set.

Problem 23. Let $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.

  1. Show that $f$ is uniformly continuous on $\mathbb{R}$.
  2. Now assume that $0<c<1$. Show that there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y)))\ldots$.
  3. Show that the result in 2. above may fail if $c=1$.

Problem 24. A function $f:\mathbb{R} \to \mathbb{R}$ is called bounded if there is an $M>0$ such that $|f(x)|\leq M$ for all $x\in\mathbb{R}$.

  1. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\mathbb{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$.
  2. Show that the result may fail without the boundedness condition.
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