Linked Questions

45
votes
6answers
7k views

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true?

Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? It seems to me like they are equal definitions in a way. Can you give me a counter-example? Thanks
37
votes
4answers
5k views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
9
votes
2answers
2k views

Construction of a function which is not the pointwise limit of a sequence of continuous functions

This is somewhat linked to a prior question of mine which was looking to see if a proof of mine regarding the Dirichlet function was correct (it wasn't). I now have an answer to the question which can ...
4
votes
2answers
2k views

Examples of nowhere continuous functions

Do you know examples of nowhere continuous functions, besides the Dirichlet function and its modifications?
5
votes
2answers
236 views

Maximum Baire class of a Riemann integrable function

In this answer (see example 1), Andrés Caicedo gives an example of a function $f : [0,1] \to \mathbb{R}$ which is Riemann integrable and is (strictly) of second Baire class. Are there Riemann ...
5
votes
1answer
202 views

Examples of functions that do not belong to any Baire class

Recently I read this post, which gives examples of Baire class 2 functions. I have also been reading the Wikipedia article on Baire functions. The article claims that "Henri Lebesgue proved that ... ...
3
votes
1answer
243 views

Is a function with a dense set of points of continuity of Baire Class 1?

Let $f:[0,1]\to\mathbb R$ be an arbitrary function. It is well known that if $f$ is of Baire Class 1, then the points at which $f$ is continuous form a dense subset of $[0,1]$. My question now is, if ...
1
vote
1answer
203 views

What is an interval contiguous to the Cantor set?

I have been reading the answer to Examples of Baire class 2 functions , and I'm trying to see what is an example of an interval contiguous to the Cantor set. Could someone lend me a hand?
4
votes
1answer
132 views

What are examples of real-valued functions of Baire class at least 3?

It's quite easy to find functions of Baire classes 0 and 1; however, the regularity of these functions falls off quite quickly, and with it, ease of construction. The most famous example of a Baire ...
1
vote
1answer
72 views

To express a integrable function as difference.

Assume $f$ is an integrable function on $[0,1]$. I want to find functions $g$ and $h$, so that $f=g-h$ almost everywhere. The functions $g$ and $h$ should be pointwise limits of continuous functions $...
0
votes
1answer
52 views

Is the zero set of the Conway base 13 function Borel?

Let $Z$ be the zero set and $S$ the set of numbers that can be written using only the digits 0 through 9 (base 13), which has measure zero. Then $Z^c\subset\mathbb{Q}+S$, which has measure zero.
2
votes
0answers
32 views

“Representative” Topics in Analysis

I have a lot of time on my hands and want to work on learning some higher level topic in analysis, but one that helps me stay in touch with (and get better) at "standard" topics in analysis. What are ...