$$ \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} $$
Due: January 19th, 2018

Math 142B Assignment 1

    1. Show that if $\mathcal{P_1}, \ldots, \mathcal{P_n}$ are partitions of the interval $[a, b]$, then there is a partition $\mathcal{P}$ of the interval $[a,b]$ which is a refinitement of every $\mathcal{P}_i$; that is, show any finite set of partitions has a common refinement.
    2. Suppose $\paren{\mathcal{P_n}}_n$ is an infinite sequence of partitions of $[a, b]$. Must there be a partition which is a refinement of every partition in the sequence? (Make sure to check the definition of "partition" carefully.)
  1. Let $f : [0, 2] \to \R$ be defined by $f(t) = t$. Find a partition $\mathcal{P}$ of $[0, 2]$ such that $$U(f, \mathcal{P}) - L(f, \mathcal{P}) \lt \frac{1}{256}.$$
    1. Suppose that $f : [a,b] \to \R$ is integrable, and $g : [a,b]\to\R$ is such that $g(x) = f(x)$ for all $x \in [a,b] \setminus\set{x_0}$. Show that $g$ is integrable, and $$\int_a^b f(t)\,dt = \int_a^b g(t)\,dt.$$
    2. Using the previous part, show that if $f : [a,b] \to \R$ is integrable and $g : [a,b]\to\R$ is such that $g(x) = f(x)$ for all but finitely many points in $[a, b]$ then $g$ is integrable, and $$\int_a^b f(t)\,dt = \int_a^b g(t)\,dt.$$ (Hint: use induction.)
    3. Suppose that $f : [a, b] \to \R$ is integrable, and $g : [a,b] \to \R$ is such that $g(x) = f(x)$ except for at a sequence of points $(x_n)_n$. Must $g$ be integrable? Prove it, or provide a counterexample.
  2. Suppose $f : [a,b] \to \R$ is bounded, $c \in (a, b)$, and $f$ is integrable on both $[a, c]$ and $[c, b]$. Show that $f$ is integrable on $[a, b]$.
    1. Find bounded functions $f, g : [0, 1] \to \R$ so that $f + g$ is integrable, but $f$ is not.
    2. Find bounded functions $f, g : [0, 1] \to (0, \infty)$ so that $fg$ is integrable, but $f$ is not.
  3. Bonus:

    Let the function $f$ be defined as follows: $$\begin{align*} f : [0,1] &\longrightarrow \R\\ x &\longmapsto \begin{cases}\frac1q & \text{ if } x = \frac{p}{q} \in \Q \text{ with } p, q \in \N, \gcd(p,q) = 1{\color{red},} \text{ and } x \neq 0\\ 0 & \text{ otherwise.} \end{cases} \end{align*}$$ Show that $f$ is integrable, and find its integral.

    Notice that $\frac{d}{dt}\int_0^t f(x)\,dx \neq f(t)$.