CRN 11378: HW 4

Revision as of 13:08, 12 November 2019

Problem 25. Let $f:[a,b]\to\mathbb{R}$ be an increasing function. Show that $\lim_{x\to a}f(x)$ exists. What can you say about the relationship between this limit and $f(a)$ ?

Problem 26. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$ . Show that $h$ is continuous on $\mathbb{R}$ .

Problem 27. Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ , and assume that for all $\varepsilon>0$ there is an $N>0$ such that $|f(x)|<\varepsilon$ for all $x$ satisfying $|x|>N$ . Show that $f$ is uniformly continuous on $\mathbb{R}$ .

Problem 28. Let $f:[a,b]\to\mathbb{R}$ be a function. We say $f$ satisfies $(*)$ if there is an $M>0$ such that $|f(x)-f(y)|\leq M\cdot |x-y|$ for all $x,y\in [a,b]$ .

Let $g:[0,1]\to\mathbb{R}$ be given by $g(x)=\sqrt{x}$ . Show that $g$ does not satisfy $(*)$ .
Is $g$ uniformly continuous on $[a,b]$ ? Is $g$ uniformly continuous on $(a,b)$ ? Explain!

@@ Line 1: / Line 1: @@
-'''Problem 16.'''
+'''Problem 25.'''
-# Show: If $x$ is an accumulation point of $A\cup B $, then$ x $is an accumulation point of$ A $, or$ x$ is an accumulation point of $B$ (or both).
+Let $f:[a,b]\to\mathbb{R}$ be an increasing function. Show that $\lim_{x\to a}f(x)$ exists. What can you say about the relationship between this limit and $f(a)$?
-# Does the result also hold for a countably infinite collection of sets? Give a proof, or provide a counterexample.
-'''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 $F$.
+'''Problem 26.'''
+Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that  $h$  is continuous on $\mathbb{R}$.
-'''Problem 18.''' Find all accumulation points of the set
+'''Problem 27.'''
-\[\left\{\frac{1}{m}+\frac{1}{n}\ |\ m,n\in\mathbb{N}\right\}\]
+Let $f:\mathbb{R}\to\mathbb{R} $be continuous on$ \mathbb{R}$, and assume that for all $\varepsilon>0 $there is an$ N>0 $such that$ |f(x)|<\varepsilon $for all$ x $satisfying$ |x|>N $. Show that$ f $is uniformly continuous on$ \mathbb{R}$.
-Remember that $A=B\  \Leftrightarrow\  (A\subseteq B)\wedge (B\subseteq A)$.
-'''Problem 19.''' Show: If $X\subseteq \mathbb{R}$ is both open and closed, then $X=\mathbb{R}$ or $X=\emptyset$.
+'''Problem 28.'''
+Let $f:[a,b]\to\mathbb{R}$ be a function. We say  $f$  satisfies  $(*)$  if there is an $M>0 $such that$ |f(x)-f(y)|\leq M\cdot |x-y|$ for all $x,y\in [a,b]$.
-'''Problem 20.''' Consider the following sets:
+# Let $g:[0,1]\to\mathbb{R} $be given by$ g(x)=\sqrt{x} $. Show that$ g $does not satisfy$ (*)$.
-\[A=\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4}\ldots\right\},\quad B=\left\{1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}\ldots\right\}, \quad C=\mathbb{Q}\cap[0,1]\]
+# Is  $g$  uniformly continuous on $[a,b] $? Is$ g $uniformly continuous on$ (a,b)$? Explain!
-For the sets that are compact, explain why. For the other ones, show that they have an open cover without finite subcover.

CRN 11378: HW 4

Revision as of 13:08, 12 November 2019

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