Posts Tagged ‘continuous’

the Cantor function and some of its properties

December 16, 2013 Leave a comment

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.

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There isn’t such a function

February 9, 2012 2 comments

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}.

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Categories: Analysis, Olympiad Tags: ,

Useful continuity

February 18, 2011 Leave a comment

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

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Functional is continuous iff kernel is closed

January 27, 2011 Leave a comment

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.

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Characterisation of continuous functions

May 22, 2010 Leave a comment

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.

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Reccurent function sequence SEEMOUS 2010

April 26, 2010 Leave a comment

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

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Hahn-Banach application.

January 19, 2010 Leave a comment

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.

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