Ian Charlesworth

Math 104 - Introduction to Analysis

Some information about the final exam

The final exam will consist of 12 problems. The first 6 will be worth 5 points each, and short; the remaining 6 will be worth 10 points each, and longer. Below I have produced a list of some problems I consider short and long; to make a practice final, select 6 of each type at random.

Short problems will involve stating definitions or theorems, or making one- or two-sentence arguments. Some examples:

• Define {open, closed, compact, continuous, uniformly continuous, differentiable at $x_0$, separates points, vanishes nowhere, Cauchy sequence, uniformly convergent, pointwise convergent, monotonically increasing sequence, montonically increasing function, least upper bound property, least upper bound, greatest lower bound, complete, limit point, interior point, $\displaystyle\lim_{x\to\infty}f(x)$, $\displaystyle\lim_{x\to-\infty}f(x)$, $\displaystyle\lim_{x\to x_0} f(x) = \infty$, $\displaystyle\lim_{x\to x_0}f(x) = -\infty$, partition, ...}.
• State the three properties a function $d: M\times M \to \R_{\geq0}$ must satisfy to be a metric.
• State the three properties a relation $\sim$ on a set $B$ must satisfy to be an equivalence relation.
• State the three properties a relation $\prec$ on a set $B$ must satisfy to be an order.
• State the {mean value theorem, extreme value theorem for real-valued functions, intermediate value theorem for real-valued functions, Heine-Borel theorem, monotone convergence theorem, sequential characterisation of a limit, sequential characterisation of continuity, Stone's theorem, Stone-Weierstrass theorem, ...}.
• Does there exist a continuous onto function $f: [0, 1] \to \R$? Provide an example or explain why no example exists.
• The algebra $\mathcal{C}([0, 1]) = \set{f : [0, 1] \to \R \mid f \text{ is continuous}}$ is not a field. State at least one of the properties of fields which it fails to satisfy.
• Describe all continuous functions from $\R$ to $\Z$. Justify briefly.
• Let $S$ be a set and $d_{\text{discrete}}$ the discrete metric on $S$: that is, $d_{\text{discrete}}(x, y) = 1$ whenever $x \neq y$. Which subsets of $S$ are {closed, open, compact}? Justify briefly.

The remaining 6 will be worth 10 points each, and longer. These will include proofs which have been presented in class or the homework, and thematically similar but new problems. Below are several examples; some of these were rejected from the actual final for being too difficult, so the average difficulty of the questions below is likely slightly higher than what will appear on the final.

• [Insert any homework problem here.]
• Prove that if $S \subseteq \R$ is bounded above, then $\sup S \in \overline{S}$.
• Prove that the continuous image of a compact set is compact.
• Prove that the continuous image of a connected set is connected.
• Prove that compact sets are bounded.
• Prove that compact sets are closed.
• Prove the monotone convergence theorem: a monotonic sequence in $\R$ converges if and only if it is bounded.
• Prove that convergent sequences are bounded.
• Prove that a function $f : X \to Y$ is continuous at $x \in X$ {if, only if} for every sequence $(a_n)_n$ converging to $x$, $(f(a_n))_n$ converges to $f(x)$.
• Prove that a set is closed {if, only if} its complement is open.
• Prove that the following two versions of "bounded" are equivalent, provided that the metric space $M$ is non-empty (that is, a set satisfies the first condition if and only if it satisfies the second):
  1. a set $S \subseteq M$ is bounded if there exists some $x \in M$ and $R \gt 0$ so that $S \subseteq B_R(x)$;
  2. a set $S \subseteq M$ is bounded if for every $x \in M$ there is some $R_x \gt 0$ so that $S \subseteq B_{R_x}(x)$.
• Prove that a subset of $\R$ is bounded in the metric space sense if and only if it is bounded above and below in the ordered set sense.
• Suppose that $S \subseteq M$ is bounded. Show that $\overline{S}$ is bounded also.
• Show that a closed subset of a compact set is compact.
• Prove that if $K$ is a compact metric space, then $K$ is complete.
• Suppose that $f : (a, b) \to \R$ is differentiable. Show that if $f(x)$ has a local maximum at $x_0$ then $f'(x_0) = 0$.
• Show that a continuous function on a compact set is uniformly continuous.
• Suppose that $(a_n)_n$ and $(b_n)_n$ are sequences in a metric space $(M, d)$ which both converge to $x \in M$. Suppose that $(c_n)_n$ is a sequence in $M$ with the property that for every $n \in \N$, \[c_n \in \set{a_k \mid k \geq n} \cup \set{b_k \mid k \geq n}.\] Show that $(c_n)_n$ converges to $x$.
• Prove the following step of the Mean Value Theorem: supposing that $f, g : [a, b] \to \R$ are continuous functions differentiable on $(a, b)$ and $h(t) = (f(b)-f(a))g(t) - (g(b)-g(a))f(t)$, there is some $t \in (a, b)$ with $h'(t) = 0$.
• Suppose that $f : (a, b) \to \R$ is differentiable, and $f'(x) \gt 0$ for all $x \in (a, b)$. Prove that $f$ is monotonically increasing.
• Suppose that $g : \R \to\R$ is differentiable, $a \lt b$, and $g'(a) \lt y \lt g'(b)$ for some $y$. Show that there is $x \in (a, b)$ so that $g'(x) = y$.
• Suppose $f : [0, 1] \to \R$ is continuous and one-to-one. Prove that $f$ is monotonic.
• Define a sequence $(f_n)_n$ of functions $[0, 1] \to \R$ as follows: given $n \in \N$, let $k_n, m_n \in \N \cup\set0$ be such that $m_n \lt 2^{k_n}$ and $n = m_n + 2^{k_n}$; then, define \[f_n(x) = \begin{cases} 1 & \text{ if } x\in \left[\frac{m}{2^k}, \frac{m+1}{2^k}\right] \\ 0 & \text{ else.}\end{cases}\] Show that $(f_n)_n$ does not converge pointwise, but \[\lim_{n\to\infty}\int_0^1 f_n(t)^2\,dt = 0.\]
• Suppose that $g : X \to Y$ is uniformly continuous and $(f_n)_n$ is a sequence of functions $M \to X$ converging uniformly to $f$. Prove that $g \circ f_n$ converges uniformly to $g\circ f$.
• Suppose that $f : \R \to \R$ is differentiable at $0$. Let $g : \R\to\R$ be defined by $g(x) = f(\abs{x})$. Prove that $g$ is differentiable at $0$ if and only if $f'(0) = 0$.
• Suppose that $(a_n)_n$ is a sequence in $\R$. Define $(m_n)_n$ by \[m_n = \sup\set{a_k \mid k \geq n}.\] Prove that $(m_n)_n$ converges in the extended real numbers (i.e., $(m_n)_n$ either converges, diverges to $\infty$, or diverges to $-\infty$).
• Suppose that $f : \R^k \to \R^m$ is continuous and has the property that for any compact set $K \subseteq \R^m$, $f^{-1}(K) \subseteq \R^k$ is compact. Prove that $f(C)$ is closed whenever $C \subseteq \R^k$ is closed.
• Let $f : [1, \infty) \to \R$ be given by $f(x) = \sqrt{x}$. Show that for any polynomial function $p : [1, \infty) \to \R$, $\norm{f-p}_\infty = \infty$. This means that $f$ cannot be the uniform limit of a sequence of polynomials; why does this not contradict Stone's theorem?
• Suppose that $(a_n)_n$ is a sequence in a metric space $(M, d)$. Show that the limit of $(a_n)_n$ is unique if it exists.
• Show that uniform limits of continuous functions are continuous.
• Prove the final step in the Stone-Weierstrass theorem, assuming that the previous steps are known. That is, suppose that $K$ is a compact metric space, that $\mathcal{A}$ is an algebra of continuous functions on $K$ closed under uniform convergence, that $\varphi : K \to \R$ is continuous, that $\epsilon \gt 0$, and that for every $x \in K$ there is a function $g_x \in \mathcal{A}$ so that $g_x(x) = \varphi(x)$ and $g_x(t) \gt \varphi(t)-\epsilon$ for every $t \in K$. Show that there is a function $f \in \mathcal{A}$ with \[\varphi(t)-\epsilon \lt f(t) \lt \varphi(t) + \epsilon\] for every $t \in K$.
• Suppose that $(f_n)_n$ is a sequence of integrable functions $[a, b] \to \R$ which converges uniformly to some function $f : [a, b] \to \R$. Prove that $f$ is integrable.
• Suppose that $f : [a, b] \to \R$ is integrable. Show that for every $\epsilon \gt 0$ there is a partition $\mathcal{P}$ of $[a, b]$ so that \[U(\mathcal{P}, f) - L(\mathcal{P}, f) \lt \epsilon.\]
• Suppose that $f : [a, b] \to \R$ and that $\mathcal{P}, \mathcal{Q}$ are partitions of $[a, b]$ with $\mathcal{P} \subseteq \mathcal{Q}$; that is, that $\mathcal{Q}$ refines $\mathcal{P}$. Prove that \[L(\mathcal{P}, f) \leq U(\mathcal{Q}, f).\]