# Proving Space of Differential Functions Not Closed

In summary, the conversation discusses the question of how to prove that the space of differentiable functions is not closed, but more information is needed to determine what exactly is meant by "closed" and in what context. The conversation also brings up the possibility of considering polynomials as a subset of the set of differentiable functions, but it is unclear what operation or topology is being considered.

how i prove that space of defrential function not closed?

You need to give us more details. Closed in the sense of topology or closed in the sense of being a subspace of a vector space (or a group or whatever)? Do you mean differential? What sort of objects is it acting on? Just real-valued functions of one real variable?

{x:xis polynomail} subset c[a,b]}

Still not enough information! If I were to guess, then it would be that you want to know how to prove that the space of differentiable functions between 2 topological spaces is not closed under some topology.

Is this correct? If so, which topology? Don't hold back!

I would interpret "c[a, b]" as the set of functions continuous on [a, b] but the orignal post said "differentiable" which would be "c1[a, b]".

So I take it you want to show that the set of all polynomials is not closed as a subset of the set of all differentiable functions on some interval.

But it is still not clear if you mean "closed" under some operation or "closed" in some topology. And, we would need to know what operation or topology is involved.

## 1. What is meant by "proving space of differential functions not closed"?

The space of differential functions refers to the set of all functions that can be differentiated. A space is considered closed if it contains all of its limit points. Therefore, proving that the space of differential functions is not closed means showing that there are some functions that can be differentiated, but their limits do not belong to the space.

## 2. Why is it important to prove that the space of differential functions is not closed?

Proving that the space of differential functions is not closed is important because it helps us understand the limitations of the space and the types of functions that cannot be represented by it. It also allows us to identify the areas where the space can be improved or extended to include a wider range of functions.

## 3. What are some common methods used to prove that the space of differential functions is not closed?

One common method is to use counterexamples, where a function is constructed that can be differentiated, but its limit does not belong to the space. Another method is to use the concept of completeness, where it is shown that the space is not complete and therefore cannot contain all of its limit points.

## 4. Can the space of differential functions ever be closed?

No, the space of differential functions cannot be closed. This is because there are always functions that can be differentiated, but their limits do not belong to the space. However, the space can be extended or improved to include a wider range of functions.

## 5. What are the implications of proving that the space of differential functions is not closed?

The implications of this proof vary depending on the specific context and application of the space. In general, it can help us understand the limitations of the space and the types of functions that cannot be represented by it. It can also guide future research and development of the space to make it more comprehensive and effective.