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{\bf \Large Math 112 Homework 7}\medskip

Due Tuesday April 9, 2019, on Canvas.
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You are encouraged to discuss the homework problems with 
other students. However, what you hand in should reflect your 
own understanding of the material. You are NOT allowed to copy solutions 
from other students or other sources. {\it No late homeworks will be 
accepted.}

Please pay attention to the clarity and precision of your answers. Your
solutions to the problems should always consist of carefully written 
mathematical arguments. 

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{\bf Material covered:} Rudin pages 103--111, 120--121.
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{\bf Problem 1.} (5 points) Rudin Chapter 5 Problem 2.

{\it Hint:} Recall that $g$ is defined by the equation $g(f(x))=x$.
You need to prove that, if $\{y_n\}$ is a sequence in $f((a,b))$
and $y_n\to y=f(x)$, then $\frac{g(y_n)-g(y)}{y_n-y}$ converges to
$\frac{1}{f'(x)}$. For this purpose, you will almost certainly need to
first prove that $x_n=g(y_n)$ converges to $x$, i.e.\ that $g$ is 
continuous at $y$. In order to do this, you can either try to use Theorem
4.17 (be careful!), or alternately first show that $g$ is increasing, 
and then assume that $g(y-)<g(y+)$ to derive a contradiction.
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{\bf Problem 2.} (3 points) Rudin Chapter 5 Problem 4.

{\it Hint:} Apply the mean value theorem to 
$f(x)=C_0x+\frac{C_1}{2}x^2+\dots+\frac{C_n}{n+1}x^{n+1}$.
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{\bf Problem 3.} (5 points) Rudin Chapter 5 Problem 26.
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{\bf Problem 4.} (3 points) Rudin Chapter 5 Problem 17.
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{\bf Problem 5.} (5 points) Rudin Chapter 6 Problem 2.

{\it Hint:} Argue by contradiction. Assuming $f$ is not everywhere zero,
use a suitable partition of $[a,b]$ to obtain a lower bound for the integral.
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