Continuity of functions between metric spaces #1375
+138
−1
Conversation
This file contains 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
| [metric spaces](metric-spaces.metric-spaces.md) `X` and `Y` is | ||
| {{#concept "continuous" WDID=Q170058 WD="continuous function"}} at a point `x` | ||
| if there exists a function `m : ℚ⁺ → ℚ⁺` such that whenever `x'` is in an | ||
| `m ε`-neighborhood of `x`, `f x'` is in an `ε`-neighborhood of `f x`. `m` is |
There was a problem hiding this comment.
Starting a sentence with a mathematical symbol or a code snippet leads to awkward writing, so it is advised to try and avoid it by slight reformulation, e.g., by adding a definite article.
Suggested change
| `m ε`-neighborhood of `x`, `f x'` is in an `ε`-neighborhood of `f x`. `m` is | |
| `m ε`-neighborhood of `x`, `f x'` is in an `ε`-neighborhood of `f x`. The function `m` is |
| `m ε`-neighborhood of `x`, `f x'` is in an `ε`-neighborhood of `f x`. `m` is | ||
| called the modulus of continuity of `f` at `x`. | ||
|
|
||
| `f` is called |
There was a problem hiding this comment.
Suggested change
| `f` is called | |
| The function `f` is called |
|
|
||
| `f` is called | ||
| {{#concept "uniformly continuous" WD="uniformly continuous function" WDID=Q91256217}} | ||
| if there is a single `m : ℚ⁺ → ℚ⁺` such that `f` is pointwise continuous with |
There was a problem hiding this comment.
Suggested change
| if there is a single `m : ℚ⁺ → ℚ⁺` such that `f` is pointwise continuous with | |
| if there is a single function `m : ℚ⁺ → ℚ⁺` such that `f` is pointwise continuous with |
There was a problem hiding this comment.
I suggest you add the concept of uniformly continuous functions in a second file. Also, I believe it would be slightly more conventional to call this file continuous-functions-metric-spaces. See for instance the analogous concept scott-continuous-maps-posets in domain theory.
| @@ -0,0 +1,99 @@ | |||
| # Continuity of functions between metric spaces | |||
There was a problem hiding this comment.
Suggested change
| # Continuity of functions between metric spaces | |
| # Continuous functions between metric spaces |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
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.
No description provided.