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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{Tuesday, March 10, 2025}
\newcommand{\myAssignmentTitle}{Homework Assignment 5}
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\begin{document}

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\begin{questions}
    
\question Prove that a polynomial $p(x)$ with odd degree has a least one real zero. 

\bigskip

\question Suppose that the function \ $f : [a, \ b] \to \mathbb{R}$ \ is continuous.  Show that for any \ $n \in \mathbb{N}$ \ and points

$x_1, x_n, \ldots, x_n \in [a, \ b]$, \  there is a point \ $z \in [a, \ b]$ \ such that \ $\displaystyle f(z) = \frac{f(x_1) + f(x_2) + \cdots + f(x_n)}{n}$.

\bigskip 

\question Suppose that the function \ $f : \left[0, \ 1\right] \to \mathbb{R}$ \ is continuous with \ $f(0) > 0$, \ and \ $f(1) = 0$. \ Show that there is a number \ $x_0 \in \left(0, \ 1\right]$ \ such that \ $f(x_0) = 0$ \ and \ $f(x) > 0$ \ for every \ $x \in \left[0, \ x_0\right)$.

\bigskip

\question Let $f(x) = \begin{cases} 0 & \text{if $x=0$}, \\ \sin\left(\frac{1}{x}\right) & \text{otherwise}. \end{cases}$

Show that $f$ has the intermediate value property on all of $\mathbb{R}$.

\bigskip

%%Fitzpatrick 3.5.8
\question Suppose that a continuous function \ $f : \mathbb{R} \to \mathbb{R}$ \ is periodic; that is, there is a number \ $p > 0$ \ such that \ $f(x + p) = f(x)$ \ for all $x \in \mathbb{R}$. \ Show that \ $f : \mathbb{R} \to \mathbb{R}$ \ is uniformly continuous.

\bigskip

%%19.4
\question Prove that if $f$ is uniformly continuous on a bounded set $S$, then $f$ is a bounded function on~$S$.

\bigskip

%%19.7
\question Let $f$ be a continuous function on $[0, \, \infty)$.  Prove that if $f$ is uniformly continuous on $[k, \, \infty)$ for some $k > 0$, then $f$ is uniformly continuous on $[0, \, \infty)$.

\bigskip

%%19.12
\question Let $f$ be a continuous function on $[a, \, b]$.  Show that the function $f^*$ defined by
\begin{displaymath} f^*(x) = \sup \left\{f(y) \mid a \leq y \leq x \right\}, \text{ for } x \in [a, \, b], \end{displaymath}
is an increasing continuous function on $[a, \, b]$.

\bigskip

%%20.16
\question Suppose the limits \ $\displaystyle L_1 = \lim_{x \to a^+} f_1(x)$ \ and \ $\displaystyle L_2 = \lim_{x \to a^+} f_2(x)$ \ both exist.
\begin{parts}
\part Show that if \ $f_1(x) \leq f_2(x)$ \ for all $x$ in some interval \ $(a, \, b)$, \ then \ $L_1 \leq L_2$.
\part Suppose \ $f_1(x) < f_2(x)$ \ for all $x$ in some interval \ $(a, \, b)$. \ Can you conclude that \ $L_1 < L_2$?
\end{parts}

\bigskip

%%20.17
\question Show that if \ $\displaystyle \lim_{x \to a^+} f_1(x) = \lim_{x \to a^+} f_3(x) = L$ \ and if \ $f_1(x) \leq f_2(x) \leq f_3(x)$ \ for all $x$ in some

interval \ $(a, \, b)$, \ then \ $\displaystyle \lim_{x \to a^+} f_2(x) = L$.  (Note: Be sure to show that \ $\displaystyle \lim_{x \to a^+} f_2(x)$ \ exists, since this is not assumed.)

\end{questions}

\end{document}