Problem 1.

Problem 1 (5 points) Let $A \subset \mathbb{R}$ be a bounded non-empty set, and let $B:={|x|: x \in A}$. Prove that
$$\sup B-\inf B \leq \sup A-\inf A$$
(Hint: Find some $|x|$ close to $\sup B$ and some $|y|$ close to $\inf B$, then use the reverse triangle inequality.)

Problem 2.

Problem 2 (4 points each) In this problem, we will look at some properties of the Riemann integral of the absolute value of a function.
(a) Suppose $f \in \mathscr{R}[a, b]$. Show that $|f| \in \mathscr{R}[a, b]$, and that
$$0 \leq\left|\int_a^b f\right| \leq \int_a^b|f|$$
(b) Find an example of a function $f:[a, b] \rightarrow \mathbb{R}$ such that $|f| \in \mathscr{R}[a, b]$ but $f \notin \mathscr{R}[a, b]$.
(Hint: Think about the Dirichlet function)
(c) Suppose $f:[a, b] \rightarrow \mathbb{R}$ is continuous. Show that if $f(c)>0$ for some $c \in[a, b]$, then there exists some $\delta>0$ such that
$$\int_{c-\delta}^{c+\delta} f>0$$
(Remark: Try to be careful about strict/non-strict inequalities and open/closed intervals in this part.)
(d) Suppose $f:[a, b] \rightarrow \mathbb{R}$ is continuous. Show that
$$\int_a^b|f|=0$$
if and only if $f(x)=0$ for all $x \in[a, b]$

Problem 3.

Problem 3 (5 points) Suppose $F$ and $G$ are continuously differentiable functions defined on $[a, b]$ such that $F^{\prime}(x)=G^{\prime}(x)$ for all $x \in[a, b]$. Using the fundamental theorem of calculus, show that $F$ and $G$ differ by a constant. That is, show that there exists a $C \in \mathbb{R}$ such that $F(x)-G(x)=C$
(Remark: This is justifying the “rule” of adding a constant $\int f+C$ to indefinite integration when you are computing an antiderivative. Make sure to use the right form of the fundamental theorem of calculus.)

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