Continuity of a given map

In summary: So phi is not injective.In summary, the map phi : C -> I is a continuous map that maps each point of the middle third Cantor set C to the set of real numbers I=[0,1] according to the rule: 0.a_1a_2a_3... -> 0.b_1b_2b_3... where b_i = a_i / 2. This can be proven by showing that phi is continuous at each point x in the domain using the epsilon-delta definition of continuity. To prove that phi is not bijective, we can use the fact that certain numbers in C have multiple ternary representations, leading to the same image in I.
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Homework Statement


Consider the map phi : C -> I which maps each point of the middle third Cantor set C, considered as a subset of real numbers between 0 and 1 written in base 3 and containing only digits 0 and 2, to the set of real numbers I=[0,1] written in base 2, according to the rule: 0.a_1a_2a_3... -> 0.b_1b_2b_3... where b_i = a_i / 2

(1) Prove that phi is a continuous map of C onto I.
(2) Prove that phi is not bijective.

Homework Equations


The Attempt at a Solution


I decided to use the definition of continuity that all open sets in I must have open pre-images in C. I tried saying pick (1/2,1) in I (that is, all elements of the form 0.1b_2b_3...). This open intervals pre-image in C would be the intersection of (2/3,1) with the cantor set, C.

It's really sketchy in my head and I would love some help. Also proving that phi is not bijective. I feel it may have something to do with certain decimal expansions.

Thanks.
 
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  • #2
There are more elegant ways to do this, but I would just get my hands dirty and prove phi is continuous at each x in the domain, the boring epsilon-delta way.

Here is a hint, by looking at how to prove phi is continuous at x=1/4. Note 1/4 has ternary expansion 0.020202... , so phi(1/4)=0.010101... in binary, i.e. 1/3.

Let epsilon > 0.

Find N such that 1/2^N < epsilon. For this hint, suppose N=5 works.

(Note: I'm not going to be careful and figure out whether I meant N or N-1 or N+1 or N+2 in this hint.)

So go out to the fifth (or sixth) binary "decimal" place in f(1/4)=1/3, so you have 0.010101..., and furthermore L < 1/3 < U, where L=0.01010100000000... and U= 0.01010111111111... (binary).

Note U-L<1/2^5 (or 6 or whatever).

Now L and U are the images of 0.020202000000... and 0.0202022222222... (ternary) respectively, and these two numbers differ by 0.00000022222... (ternary) which is 1/3^5 (or 6 or 7), so now you have your delta (or divide it by 2).

Just write this up in general.

You may or may not have to make a special case if x ends with 0 repeating or 2 repeating (ternary). For example 1/3 = 0.100000... = 0.0222222... (ternary) so 1/3 is in C. What is phi(1/3)? Also 2/3 = 0.12222222... = 0.2000000... (ternary) so 2/3 is in C. What is phi(2/3)?
 

1. What is the definition of continuity for a given map?

Continuity of a given map refers to the property of a function or mapping where small changes in the input result in small changes in the output. In other words, the function does not have any abrupt changes or jumps in its values.

2. How is continuity of a given map mathematically defined?

Mathematically, a function f is continuous at a point a if the limit of f(x) as x approaches a exists and is equal to f(a). This means that the value of the function at a point a is equal to the value of the limit as x approaches a.

3. What are the three types of continuity for a given map?

The three types of continuity are pointwise continuity, uniform continuity, and global continuity. Pointwise continuity means that each point in the domain has a corresponding point in the range that is close to it. Uniform continuity means that there is a single delta that works for all points in the domain. Global continuity refers to the continuity of a function over its entire domain.

4. How can you determine if a given map is continuous?

To determine if a given map is continuous, you can use the definition of continuity and check if the limit of the function exists at each point in the domain. You can also use theorems and properties of continuous functions, such as the intermediate value theorem and the composition of continuous functions theorem.

5. What are some real-world applications of continuity of a given map?

Continuity of a given map is essential in various fields such as physics, engineering, and economics. In physics, it is used to describe the motion of objects and the flow of fluids. In engineering, it is used to design structures and systems that can withstand continuous loads. In economics, it is used to model and predict the behavior of markets and financial systems.

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