### Archive

Posts Tagged ‘continuous’

## the Cantor function and some of its properties

Let’s start by definining the Cantor set. Define ${C_0=[0,1]}$ and ${C_{n+1} = C_n/3 \cup (2/3+C_n/3)}$. At each step we delete the middle third of all the intervals of ${C_n}$ to obtain ${C_{n+1}}$. Note that we obviously have ${C_{n+1} \subset C_{n}}$ (an easy inductive argument) and ${|C_n|=(2/3)^n}$. The sets ${C_n}$ are compact and descending, therefore we can define ${C=\bigcap_{n=0}^\infty C_n}$ which is a compact subset of ${[0,1]}$ with zero measure and it is called the Cantor set.

Since at each step we remove a middle third of all the intervals in ${C_n}$, one way to look at the Cantor set is to look at the ternary representation of the points in it. In the first step, we remove all the elements of ${[0,1]}$ which have ${1}$ on their first position in the ternary representation. In the second step we remove those (remaining) which have ${1}$ on the second position, and so on. In the end we are left only with elements of ${[0,1]}$ which have only digits ${0,2}$ in their ternary representation. Using this representation we can construct a bijection between ${C}$ and ${[0,1]}$ which maps

$\displaystyle x=\sum_{n=1}^\infty \frac{a_n}{3^n} \mapsto \sum_{n=1}^\infty \frac{b_n}{2^n}$

where ${b_n=0}$ if ${a_n=0}$ and ${b_n=1}$ if ${a_n=2}$. This proves that the Cantor set is uncountable.

We can construct the Cantor function ${g:[0,1] \rightarrow [0,1]}$ in the following way. Denote ${R_n}$ the set ${C_n\setminus C_{n+1}}$ (i.e. the set removed in step ${n}$). On ${R_1}$ we let ${g(x)=1/2}$. On ${R_2}$ we have two intervals: on the left one we let ${g(x)=1/4}$ and on the right one we let ${g(x)=3/4}$. We continue like this iteratively, at each step choosing ${g}$ constant on each of the intervals which construct ${R_n}$ such that the constant on an interval is the mean of the values of neighboring interval values.

## There isn’t such a function

Prove that there is no continuous function $f: \Bbb{R} \to \Bbb{R}$ such that $f(\Bbb{Q}) \subset \Bbb{R}\setminus \Bbb{Q}$ and $f(\Bbb{R}\setminus \Bbb{Q}) \subset \Bbb{Q}$.

## Useful continuity

Suppose $f: \Bbb{R}^p \to \Bbb{R}$ is integrable and bounded. Prove that the mapping
$(\Bbb{R}^p)^n \ni (a_1,a_2,...,a_n) \mapsto \displaystyle \int_{\Bbb{R}^p}f(x+a_1)f(x+a_2)...f(x+a_n)f(x)dx$ is continuous.

Then use this result to solve the following problem

## Functional is continuous iff kernel is closed

Let $f$ be a linear functional on a topological vector space $X$. Assume $f(x)\neq 0$ for some $x \in X$. Then the following properties are equivalent:

1. $f$ is continuous
2. $\ker f=\{x : f(x)=0\}$ is closed
3. $\ker f$ is not dense in $X$
4. $f$ is bounded in some neighborhood of $0$.

## Characterisation of continuous functions

Let ${f:\Bbb{R}^n \rightarrow \Bbb{R}^m}$ be a map such that the image of any compact set is compact, and the image of any connected set is connected. Prove that ${f}$ is continuous.

## Reccurent function sequence SEEMOUS 2010

Suppose $f_0:[0,1]\to \mathbb{R}$ is a continuous function, and define the sequence $(f_n)_n, f_n:[0,1]\to \mathbb{R}$ in the following way:

$f_n(x)=\int_0^x f_{n-1}(t)dt,\ \forall x \in [o,1]$.

a) Prove that the series $\sum_{n\geq 0}f_n(x)$ converges for any $x \in [0,1]$.

b) Find an explicit formula in terms of $x$ for the above series.

Seemous 2010, Problem 1

## Hahn-Banach application.

Suppose $X$ is a normed space and $X_0$ is a closed subspace of $X$ and $x_0 \in X \setminus X_0$. Then we can find $f \in X^\prime$ such that $f(x_0)=1$ and $f(x)=0,\ \forall x \in X_0$.