## the Cantor function and some of its properties

Let’s start by definining the Cantor set. Define and . At each step we delete the middle third of all the intervals of to obtain . Note that we obviously have (an easy inductive argument) and . The sets are compact and descending, therefore we can define which is a compact subset of with zero measure and it is called the *Cantor set*.

Since at each step we remove a middle third of all the intervals in , 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 which have on their first position in the ternary representation. In the second step we remove those (remaining) which have on the second position, and so on. In the end we are left only with elements of which have only digits in their ternary representation. Using this representation we can construct a bijection between and which maps

where if and if . This proves that the Cantor set is uncountable.

We can construct the Cantor function in the following way. Denote the set (i.e. the set removed in step ). On we let . On we have two intervals: on the left one we let and on the right one we let . We continue like this iteratively, at each step choosing constant on each of the intervals which construct 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 such that and .

## Useful continuity

Suppose is integrable and bounded. Prove that the mapping

is continuous.

Then use this result to solve the following problem

## Functional is continuous iff kernel is closed

Let be a linear functional on a topological vector space . Assume for some . Then the following properties are equivalent:

- is continuous
- is closed
- is not dense in
- is bounded in some neighborhood of .

## Characterisation of continuous functions

Let be a map such that the image of any compact set is compact, and the image of any connected set is connected. Prove that is continuous.

## Reccurent function sequence SEEMOUS 2010

Suppose is a continuous function, and define the sequence in the following way:

.

a) Prove that the series converges for any .

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

*Seemous 2010, Problem 1*

## Hahn-Banach application.

Suppose is a normed space and is a closed subspace of and . Then we can find such that and .