# Can Continuous Functions on [0,1] Approach 1 but Not Equal 1 at x=1?

• Ed Quanta
In summary, the author is trying to show that D does not contain all of its limit points by finding a function that does not exist in D.
Ed Quanta
So if D={f is an element of C[0,1];f(1) does not equal 1}

and C[0,1] is the set of complex valued continuous functions on the interval [0,1], is there a function f such that f approaches 1 and f(1) does not equal 1?

I feel like there has to be one but I am unable to construct one since if the lim as x approaches 1 does not equal f(1)=1, then f wouldn't be continuous right?

I'm trying to show that D doesn't contain all of its limit points since that would be all that is required to show D is not closed.

Help with finding this function if there is one please.

Ed Quanta said:
So if D={f is an element of C[0,1];f(1) does not equal 1}

and C[0,1] is the set of complex valued continuous functions on the interval [0,1], is there a function f such that f approaches 1 and f(1) does not equal 1?

I feel like there has to be one but I am unable to construct one since if the lim as x approaches 1 does not equal f(1)=1, then f wouldn't be continuous right?

I'm trying to show that D doesn't contain all of its limit points since that would be all that is required to show D is not closed.

Help with finding this function if there is one please.

I don't understand what you are looking for. If you're trying to show that D is not closed by finding a limit point outside of D, then all you have to do is find a sequence of functions in D that converges to a function not in D.

For example, take f_n(x) = n / (n + 1). Then every f_n is in D, but the sequence converges to f(x) = 1, which is not in D.

Yeah, you basically answered my question

Ed Quanta said:
Yeah, you basically answered my question

But then why were you asking about trying to find a function such that f(1) != 1 but f(x) -> 1 as x -> 1?

## 1. What is C[0,1] in the context of mathematics?

C[0,1] is a notation used in mathematics to represent the set of continuous functions defined on the interval [0,1]. This means that the function is defined and has a value for every point between 0 and 1, and there are no breaks or jumps in the graph of the function.

## 2. How is C[0,1] different from other function spaces?

C[0,1] is a specific function space that includes only continuous functions on the interval [0,1]. Other function spaces may include different types of functions, such as differentiable functions or integrable functions, and may have different intervals or domains.

## 3. What are some common uses of C[0,1] in mathematics?

C[0,1] is commonly used in analysis and topology to study the properties of continuous functions. It is also used in differential equations and functional analysis to model real-world phenomena and solve mathematical problems.

## 4. Can all functions on the interval [0,1] be represented by C[0,1]?

No, not all functions on the interval [0,1] are continuous and therefore cannot be represented by C[0,1]. For example, step functions and functions with discontinuities cannot be represented in this function space.

## 5. Are there any limitations to using C[0,1] in mathematical analysis?

While C[0,1] is a useful function space, it does have some limitations. For example, it cannot accurately represent functions with infinitely many points of discontinuity, and it may not be able to capture the behavior of certain highly irregular functions. In these cases, other function spaces may be more appropriate.

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