Proving Continuity of Real-Valued Functions Using G_delta Sets

  • Thread starter fourier jr
  • Start date
  • Tags
    Sets
In summary, the set of points at which a real-valued function is continuous is a subset of the set of points at which the function is defined.
  • #1
fourier jr
765
13
"Let f be a real-valued function defined for all real numbers. Prove that the set of points at which f is continuous is a [tex]G_\delta[/tex]."
(a set is a [tex]G_\delta[/tex] if it is the intersection of a countable collection of open sets)

I think it's obvious that I should use the topological/open-set definition of continuous, and then intersect a bunch of open sets but I'm not sure how to write it down. (if that makes any sense)
 
Physics news on Phys.org
  • #2
Ahh, question 53 from Royden..taking math 435? (I went to UVic years ago).

f is continuous at a point if for every epsilon there is a delta..blah blah, you know the rest. For a hint, try to make the "for every" part handled by the intersection of your sets, that is take your sequence of sets to characterize the points were f satisfies a weaker condition than full blown continuity, but one that is getting "closer" to the full definition as you move along your sequence of sets.
 
  • #3
fourier jr said:
"Let f be a real-valued function defined for all real numbers. Prove that the set of points at which f is continuous is a [tex]G_\delta[/tex]."
(a set is a [tex]G_\delta[/tex] if it is the intersection of a countable collection of open sets)

I think it's obvious that I should use the topological/open-set definition of continuous, and then intersect a bunch of open sets but I'm not sure how to write it down. (if that makes any sense)

Well, in topology you're going to want to take unions, not intersections of open sets. That said, I think you'll be better off if you work with neighborhoods rather than abstract topology definitions.
 
  • #4
Hrm, I spent a couple minutes thinking about the problem, and came up with the same approach as shmoe, so it must be right. :biggrin:


Nate, since he's after an element of [itex]G_{\delta}[/itex], and not an open set, it's okay to intersect his open sets. In fact, the theorem couldn't be true using just open sets, because there are functions continuous at exactly one point!
 
  • #5
shmoe said:
..taking math 435?

yup... can't wait to get to the riesz-fischer theorem & related stuff (because I'm Fourier jr)



thanks for the tips everybody; i think it worked out ok. :smile:
 

1. What is a set?

A set is a collection of distinct objects or elements, with no consideration given to the order or repetition of the elements.

2. What is the cardinality of a set?

The cardinality of a set is the number of elements in the set. It is often denoted by the symbol |A|, where A is the set.

3. How are sets represented in mathematics?

In mathematics, sets are often represented using curly braces { } and listing the elements of the set separated by commas. For example, the set of even numbers can be represented as {2, 4, 6, 8, ...}.

4. What is the difference between a subset and a proper subset?

A subset is a set that contains all the elements of another set. A proper subset is a subset that contains some, but not all, of the elements of another set.

5. Can sets contain other sets?

Yes, sets can contain other sets. These are known as nested sets or subsets. For example, the set of all prime numbers can be considered a subset of the set of all natural numbers.

Similar threads

Replies
2
Views
294
Replies
2
Views
265
  • Topology and Analysis
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
455
  • Topology and Analysis
Replies
21
Views
2K
  • Calculus and Beyond Homework Help
Replies
22
Views
216
  • Topology and Analysis
Replies
7
Views
3K
Replies
1
Views
829
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Back
Top