Math 104 Assignment 11

1. A pathological function

   Let $f : [0,1] \to \R$ be defined as follows: \[ f(x) = \begin{cases} 0 & \text{ if } x \notin \Q \\ 1 & \text{ if } x = 0 \\ \frac1q & \text{ if } x = \frac{p}{q} \text{ with } p \in \Z, q \in \N, \text{ and } p, q \text{ have no common factor}.\end{cases}\] (For example, $f(0) = f(1) = 1$, $f(1/2) = 1/2$, $f(1/4) = f(3/4) = 1/4$.)

   Prove that $f$ is integrable, and find its integral.

2. Integrals are insensitive to individual points

   1. Suppose that $f : [a, b] \to \R$ is bounded, and continuous except at finitely many points. Prove that $f$ is integrable.
   2. Suppose that $f : [a, b] \to \R$ is integrable, and $g : [a, b] \to \R$ is such that $f(x) = g(x)$ for all but finitely many $x \in [a, b]$. Show that $g$ is integrable, and \[\int_a^b f(x)\,dx = \int_a^b g(x)\,dx.\]

3. Uniform convergence and derivatives

   For each $n \in \N$, let \begin{align*} f_n : \R &\longrightarrow \R \\ x &\longmapsto \frac{x}{1+nx^2}. \end{align*} Show that the sequence $(f_n)_n$ converges uniformly (on $\R$) to some function $f$, and that $(f_n')_n$ converges to $f'$ pointwise on $\R\setminus\set0$, but not at $0$.

4. Uniform convergence and boundedness

   Suppose that $(f_n)_n$ is a sequence of functions with $f_n : M \to \R$, so that each $f_n$ is bounded. Suppose further that $(f_n)_n$ converges uniformly to some $f : M \to \R$. Show that $(f_n)_n$ is uniformly bounded: that is, there is some $T \in \R$ so that for all $x \in M$ and all $n \in \N$, \[\abs{f_n(x)} \leq T.\]

5. Functions orthogonal to polynomials are zero

   Let $f : [0, 1] \to \R$ be continuous.
   1. Show that $f \equiv 0$ if \[\int_0^1 f(t)^2\,dt = 0.\](The notation $f \equiv 0$ means $f(t) = 0$ for all $t$ in the domain of $f$.)
   2. Suppose that for every $n \in \N\cup\set0$, \[\int_0^1 f(t)t^n\,dt = 0.\] Show that $f \equiv 0$. (Hint: by the end of the week, we will have proved the following theorem which you may find useful: if $K \subseteq \R$ is compact and $g : K \to \R$ is continuous, there is a sequence of polynomials $(p_n)_n$ which converges uniformly to $g$ on $K$. Use this to show that $\int_0^1 f(t)^2\,dt=0$.)

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Not definitions

  1. We say a function $f : X \to Y$ is funtinuous at $p \in X$ if for every $\epsilon \gt 0$ there is some $\delta \gt 0$ so that for every $x, y \in X$ with $d_X(x, y) \lt \delta$ we have $d_Y(f(x), f(y)) \lt \epsilon$.
     1. Are all functions funtinuous?
     2. Are all continuous functions funtinuous?
     3. Are all funtinuous functions continuous?
     4. Does a funtinuous function exist?
  2. We say a function $f : X \to Y$ is protinuous if for every $x_0 \in X$, $\forall\delta \gt 0$, for all $x \in X$ with $d_X(x, x_0) \lt \delta$ there exists $\epsilon \gt 0$ so that $d_Y(f(x), f(x_0)) \lt \epsilon$.
     1. Are all functions protinuous?
     2. Are all continuous functions protinuous?
     3. Are all protinuous functions continuous?
     4. Does a protinuous function exist?
  3. We say a function $f : X \to Y$ is antitinuous at a point $p \in X$ if for every $\epsilon \gt 0$ and all $x \in X$, there exists $\delta \gt 0$ so that $d_X(x, p) \lt \delta$ implies $d_Y(f(x), f(p)) \lt \epsilon$.
     1. Are all functions antitinuous?
     2. Are all continuous functions antitinuous?
     3. Are all antitinuous functions continuous?
     4. Does a antitinuous function exist?
  4. We say a function $f : X \to Y$ is continually uniformous if for all $x, y \in X$ and all $\epsilon \gt 0$ there is a $\delta \gt 0$ so that $d_X(x, y) \lt \delta$ implies $d_Y(f(x), f(y)) \lt \epsilon$.
     1. Are all functions continually uniformous?
     2. Are all uniformly continuous functions continually uniformous?
     3. Are all continually uniformous functions uniformly continuous?
     4. Does a continually uniformous function exist?
  5. We say a function $f : X \to Y$ is continuously uniform if for every $\epsilon \gt 0$ there is a $\delta \gt 0$ so that for all $x, y \in X$, $d_Y(f(x), f(y)) \lt \epsilon$ implies $d_X(x, y) \lt \delta$.
     1. Are all functions continuously uniform?
     2. Are all uniformly continuous functions continuously uniform?
     3. Are all continuously uniform functions uniformly continuous?
     4. Does a continuously uniform function exist?

• Pointwise convergence versus metric space convergence

  Suppose $S$ is a set, and consider the set of functions from $S$ to $\R$, $\R^S = \set{f : S \to \R}$. Let us say that a metric $d$ on $\R^S$ gives rise to the topology of pointwise convergence if whenever $(f_n)_n$ is a sequence of functions in $\R^S$ we have that $(f_n)_n$ converges to a function $f$ with respect to $d$ if and only if $(f_n)_n$ converges pointwise to $f$.

  Show that there is a metric on $\R^S$ giving rise to the topology of pointwise convergence if and only if there is a one-to-one function $\varphi : S \to \N$.

• Uniform convergence versus metric space convergence

  1. Suppose that $S$ is a set and $(M, d_M)$ is a bounded metric space. Define a metric $d_\infty$ on $M^S = \set{f : S \to M}$ by \[d_\infty(f, g) = \sup_{s \in S} d_M(f(s), f(g)).\] Show that a sequence of functions $(f_n)_n$ in $M^S$ converges uniformly to $f : S \to M$ if and only if $(f_n)_n$ converges to $f$ with respect to the metric $d_\infty$.
  2. Let $(M, d_M)$ be a metric space. Define $d_M' : M \times M \to \R_{\geq 0}$ by $d_M'(x, y) = \min(1, d_M(x, y))$. Show that $d_M'$ is a metric equivalent to $d_M$ (i.e., $d_M'$ is a metric such that sets are open with respect to $d_M$ if and only if they are open with respect to $d_M'$). Notice that $(M, d_M')$ is bounded.
  3. Suppose that $X$ is a set, and $d, d'$ are equivalent metrics on $X$. Show that sequences converge with respect to $d$ if and only if they converge with respect to $d'$.
  4. Suppose that $S$ is a set and $(M, d_M)$ is an arbitrary metric space. Show that there is a metric on $M^S$ so that sequences in $M^S$ converge with respect to the metric if and only if they converge uniformly. (We say "the topology of uniform convergence is metrizable".)

• Monotone convergence of functions (Dini's Theorem)

  Suppose that $K$ is a compact metric space, and $(f_n)_n$ is a sequence of continuous functions $K \to \R$, so that $f_1 \leq f_2 \leq f_3 \leq \cdots$. Suppose further that $(f_n)_n$ converges pointwise to some continuous function $f$: that is, for every $x \in K$, \[\limni f_n(x) = f(x).\] Prove that $(f_n)_n$ converges to $f$ uniformly.

• A norm on integrable functions

  Let $R$ be the set of Riemann-integrable functions on the interval $[a, b]$. Given $f \in R$, define \[\norm{f}_2 = \paren{\int_a^b\abs{f(t)}^2\,dt}^{\frac12}.\]

  1. Show that $\norm\cdot_2$ satisfies the triangle inequality in the following sense: for all $f, g, h \in R$, \[\norm{f-h}_2 \leq \norm{f-g}_2 + \norm{g-h}_2.\]
  2. Suppose $f \in R$. Show that for any $\epsilon \gt 0$, there is a continuous function $g \in R$ with \[\norm{f-g}_2 \lt \epsilon.\] (Hint: choose a suitable partition $P = \set{x_0, \ldots, x_n}$, and take a piece-wise linear function $g$ so that $g(x_i) = f(x_i)$.)
  3. Suppose $f \in R$. Show that for any $\epsilon \gt 0$ there is a polynomial $p \in R$ so that \[\norm{f-p}_2 \lt \epsilon.\]

  The function $d_2(f, g) = \norm{f-g}_2$ on $R\times R$ is a pseudo-metric on $R$: it satisfies the triangle inequality, $d_2(f,f) = 0$, and $d_2(f, g) = d_2(g, f)$, but $d_2(f, g)=0$ does not imply $f = g$. If we define $\sim$ by $f\sim g$ whenever $d_2(f, g) = 0$, then $R/\sim$ becomes a metric space; what we have done is show that (equivalence classes of) polynomials are dense in $R/\sim$.

• Monotony

  State precisely what is meant by:
  • a monotone sequence of functions;
  • a sequence of monotone functions; and
  • a monotone sequence of monotone functions.
  Provide examples to show that these three concepts are different.

• Uniform closures of algebras need not be algebras

  Recall that for $x \in \R$, $\floor{x}$ (the floor of $x$) is the greatest integer $k \in \Z$ with $k \leq x$. For $n \in \N$, define \begin{align*} \varphi_n: \R &\longmapsto \R\\ t & \longmapsto 2^{-n}\floor{2^nt}. \end{align*} Finally, let $\mathscr{A}$ be the algebra of functions generated by $(\varphi_n)_n$. That is, $\mathscr{A}$ consists of all functions which are finite sums of scalar multiples of finite products of $\varphi_n$'s.

  Let $\overline{\mathscr{A}}$ be the uniform closure $\mathscr{A}$. Show that $\overline{\mathscr{A}}$ is not an algebra. (Hint: $x\mapsto x$ is in $\overline{\mathscr{A}}$, but $x \mapsto x^2$ is not.)

  Reflect on what this example says about trying to plot quadratic or linear functions on a pixelated display.