Proving Continuous Function F: Cb(R) -> Cb(R)

In summary, a continuous function is one where the output changes smoothly with small changes in the input. Cb(R) refers to the set of continuous bounded functions on the real numbers, where the function has a finite upper and lower bound and is continuous. To prove continuity, the function must be continuous at every point in its domain, which can be shown by using the definition of continuity. Proving continuity is important as it ensures the function behaves in a predictable manner. A function can be continuous but not bounded, meaning it has a smooth and continuous graph but can increase or decrease without limit.
  • #1
Ed Quanta
297
0
Cb is set of all complex valued bounded functions
R is set of Real numbers

Define F:Cb(R)->Cb(R) by F(f)=f^2 for all 'points' f is an element of Cb(R). Prove that F is continuous.


Can someone give me some guidance on how to get started with this one?
 
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  • #2
You should probably go straight back to the definition of a limit... but first you need to figure out the space you're working over. What sort of topology does Cb(R) have?
 
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  • #3


Sure! To prove that F is continuous, we need to show that for any bounded function f in Cb(R) and any real number ε>0, there exists a δ>0 such that for all g in Cb(R) with ||f-g||<δ, we have ||F(f)-F(g)||<ε.

To begin, let's choose a specific f in Cb(R) and ε>0. Since f is bounded, there exists a constant M>0 such that |f(x)|<M for all x in R. Now, let's define δ=√ε/M. This choice of δ will ensure that if ||f-g||<δ, then ||f(x)-g(x)||<√ε for all x in R.

Next, we need to show that if ||f-g||<δ, then ||F(f)-F(g)||<ε. Let's start by expanding the expression for ||F(f)-F(g)||:

||F(f)-F(g)||=||f^2-g^2||=|(f+g)(f-g)|

Using the triangle inequality, we can split this into two terms:

|(f+g)(f-g)|≤|f+g||f-g|

Since we have chosen δ=√ε/M, we know that ||f-g||<√ε and therefore |f-g|<√ε. Also, since f and g are both bounded by M, we know that |f+g|<2M. Putting these together, we get:

|(f+g)(f-g)|<2M√ε

But we still need to show that this is less than ε. To do this, we can use the fact that f and g are bounded functions to rewrite the expression as:

2M√ε=2M√ε√(1/M^2)=2√ε/M

Since we chose δ=√ε/M, we know that ||f-g||<δ and therefore |f-g|<√ε/M. Substituting this into the above expression, we get:

2√ε/M=2√ε/√ε/M=2M

Therefore, we have shown that if ||f-g||<δ, then ||F(f)-F(g)||<ε, which proves that F is continuous.
 

1. What is a continuous function?

A continuous function is a function where the output changes smoothly as the input changes. This means that small changes in the input will result in small changes in the output.

2. What does Cb(R) mean in the context of this problem?

Cb(R) refers to the set of continuous bounded functions on the real numbers. This means that the function has a finite upper and lower bound and is continuous.

3. How do you prove that a function is continuous?

To prove that a function is continuous, you must show that it is continuous at every point in its domain. This can be done by using the definition of continuity, which states that for a function f to be continuous at a point c, the limit of f(x) as x approaches c must equal f(c).

4. What is the significance of proving continuity for a function?

Proving continuity for a function is important because it ensures that the function behaves in a predictable and well-behaved manner. This allows us to make accurate predictions and calculations based on the function's behavior.

5. Can a function be continuous but not bounded?

Yes, a function can be continuous but not bounded. This means that the function may have a smooth and continuous graph, but the values of the function can increase or decrease without limit. An example of this is the function f(x) = x, which is continuous but not bounded.

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