Practice Test 2

There are some answers in the postscript and acrobat versions.

This test is closed book, no books, papers or notes are permitted. You may use theorems, lemmas and propositions from the class and book. Note that where $\mathbb{R}^k$ is mentioned below the standard metric is assumed.

There are 5 questions on the actual test, I think they are mostly easier than these ones.

1. Consider the function $\alpha :[0,1]\longrightarrow \mathbb{R}$ defined by

   $$\alpha (x)=\begin{cases}\frac12x&0\le x\le\frac12\\ \frac12(x+1)&\frac12\le x\le1. \end{cases}$$

   
   
   Show carefully, using results from class, that any monotonic increasing function $f:[0,1]\longrightarrow \mathbb{R}$ which is continuous at $x=\frac12$ is Riemann-Stieltjes integrable with respect to $\alpha.$
2. Let $f$ be a continuous function on $[a,b].$ Explain whether each of the following statements is always true, with brief but precise reasoning.
   1. The function $g(x)=\int_x^b f(y)dy$ is well defined.
   2. The function $g$ is continuous.
   3. The function $g$ is decreasing.
   4. The function $g$ is uniformly continuous.
   5. The function $g$ is differentiable.
   6. The derivative $g'=f$ on $[a,b].$

3. If $f:\mathbb{R}\longrightarrow \mathbb{R}$ is differntiable and satisfies $f(-10)=10,$ $f(0)=0,$ $f(10)=10$ show that there is a point where $f'(x)=1/2.$

4. If $f$ is a strictly positive continuous function on $[-1,1],$ meaning $\inf_{[-1,1]} f>0,$ show that $g(x)=\sqrt{f(x)}$ is continuous.

5. (This is basically Rudin Problem 4.14)

   Let $f:[0,1]\longrightarrow [0,1]$ be continuous.

   1. State why the the map $g(x)=f(x)-x,$ from $[0,1]$ to $\mathbb{R}$ is continuous.
   2. Using this, or otherwise, show that $L=\{x\in[0,1];f(x)\le x\}$ is closed and $\{x\in[0,1];f(x)<x\}$ is open.
   3. Show that $L$ is not empty.
   4. Suppose that $f(x)\not=x$ for all $x\in[0,1]$ and conclude that $L$ is open in $[0,1]$ and that $L\not=[0,1].$
   5. Conclude from this, or otherwise, that there must in fact be a point $x\in[0,1]$ such that $f(x)=x.$

6. Consider the function
   $$f(x)=\frac{-x(x+1)(x-100)}{x^{44}+x^{34}+1}$$
   for $x\in[0,100].$
   1. Explain why $f$ is differentiable.
   2. Compute $f'(0).$
   3. Show that there exists $\epsilon >0$ such that $f(x)>0$ for $0<x<\epsilon.$
   4. Show that there must exist a point $x$ with $f'(x)=0$ and $0<x<100.$