Every point in a proper closed interval in the real numbers is an accumulation point #1679
Conversation
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
…shake-out-signed-rational
…shake-out-signed-rational
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
Outdated
Show resolved
Hide resolved
src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
Outdated
Show resolved
Hide resolved
src/metric-spaces/limits-of-cauchy-approximations-metric-spaces.lagda.md
Outdated
Show resolved
Hide resolved
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
…paces.lagda.md Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
src/metric-spaces/accumulation-points-subsets-located-metric-spaces.lagda.md
Outdated
Show resolved
Hide resolved
fredrik-bakke
approved these changes
Nov 15, 2025
…paces.lagda.md Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
fredrik-bakke
pushed a commit
that referenced
this pull request
Nov 16, 2025
fredrik-bakke
pushed a commit
that referenced
this pull request
Nov 21, 2025
fredrik-bakke
pushed a commit
that referenced
this pull request
Nov 21, 2025
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
...and the definition of what an accumulation point is, and so on.
This is intended as a key step to #1615; let me sketch it out.
Let
fbe a function from[a,b]to the real numbers, wherea < b. The derivative of a function at a point is -- well, defined another way, but we can show that given a sequenceyapproachingxbut remaining apart from it, the derivative atxis equal to the unique limit of (f y - f x)/(y - x) as y approaches x.Such a sequence exists if and only if x is an accumulation point of the space where the function is defined. This PR proves every point in a proper closed interval is an accumulation point, so the value of any derivative of
fatxis unique, so all derivatives are homotopic, so derivatives are unique and being differentiable is a proposition.