Comments on: Sometimes all functions are continuous http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/ Mathematics for computers Wed, 20 Aug 2008 12:18:21 +0000 http://wordpress.org/?v=2.3.3 By: All functions are continuous, always @ The dreams that stuff is made of http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-9176 All functions are continuous, always @ The dreams that stuff is made of Tue, 12 Aug 2008 04:53:42 +0000 http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-9176 [...] always Posted on 2008-08-11 by Luke Palmer. Categories: Code, Haskell, Math.Dan Piponi and Andrej Bauer have written about computable reals and their relationship to continuity. Those articles [...] […] always Posted on 2008-08-11 by Luke Palmer. Categories: Code, Haskell, Math.Dan Piponi and Andrej Bauer have written about computable reals and their relationship to continuity. Those articles […]

]]> By: Constructive and Classical Mathematics < Inductio Ex Machina http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-7929 Constructive and Classical Mathematics < Inductio Ex Machina Thu, 12 Jun 2008 02:08:46 +0000 http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-7929 [...] of mathematics — denying all but the computable real numbers and functions, thereby making all functions continuous — but it is a tempting view of the [...] […] of mathematics — denying all but the computable real numbers and functions, thereby making all functions continuous — but it is a tempting view of the […]

]]> By: Andrej Bauer http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-6251 Andrej Bauer Sat, 23 Feb 2008 12:08:48 +0000 http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-6251 Reply to comment 18 by NotQuiteFunctional: I do not understand in what way `B` can be seen as a set of sets of numbers. Please explain (with examples). Reply to comment 18 by NotQuiteFunctional: I do not understand in what way `B` can be seen as a set of sets of numbers. Please explain (with examples).

]]> By: Andrej Bauer http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-6250 Andrej Bauer Sat, 23 Feb 2008 12:04:50 +0000 http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-6250 Reply to comments 16, 17 by NotQuiteFunctional: I am afraid I do not understand you very well. Reply to 16: the function you are suggesting is not computable, therefore not a function for the purposes of this discussion. Reply to 17: yes, but I suspect what you wrote is not what you meant to ask. Reply to comments 16, 17 by NotQuiteFunctional: I am afraid I do not understand you very well.

Reply to 16: the function you are suggesting is not computable, therefore not a function for the purposes of this discussion.

Reply to 17: yes, but I suspect what you wrote is not what you meant to ask.

]]> By: Andrej Bauer http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-6249 Andrej Bauer Sat, 23 Feb 2008 12:01:19 +0000 http://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/#comment-6249 Reply to comment 15 by jn: First, I did not make an argument "for continuity" of sgn, but rather "against continuity". Whether your sg is continuous (computable) depends on <a href="/2008/02/06/representations-of-uncomputable-and-uncountable-sets/" rel="nofollow">how you represent real numbers</a>. Assuming a standard representation of reals (one in which addition is computable, for example), your sg is not computable and is not continuous as a map between representations. I do not know what "two's complement representation" is for real numbers (including negative ones). Please provide more detail about your representation. Is addition computable in your representation? Reply to comment 15 by jn: First, I did not make an argument “for continuity” of sgn, but rather “against continuity”. Whether your sg is continuous (computable) depends on how you represent real numbers. Assuming a standard representation of reals (one in which addition is computable, for example), your sg is not computable and is not continuous as a map between representations. I do not know what “two’s complement representation” is for real numbers (including negative ones). Please provide more detail about your representation. Is addition computable in your representation?

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