From: hook@nas.nasa.gov (Ed Hook) Subject: Re: Nowhere-continuous function with the intermediate value property Date: 5 Jul 2000 17:57:23 GMT Newsgroups: sci.math Summary: [missing] In article , antoSPORK@mit.edu (J. Antonio Ramirez R.) writes: |> Is there such a thing? Yes. |> By the intermediate value property, I mean that if f(x)=a and f(y)=b |> for some x [x,y]. |> I am guessing it can be constructed using "condensation of |> singularities" from the function sin(1/x), in the same manner that a |> nowhere-differentiable function can be constructed with this method |> from the absolute value function. I can elaborate on request, but if |> someone knows what I'm talking about, I would be grateful to hear |> whether the idea works. I'll outline the construction of an example, but it won't look like the approach that you're suggesting *exactly*.(It's in the spirit of "condensation of singularities", but it piles on the weirdness in disjoint chunks ... ) Start with the usual Cantor set, C, contained in [0,1]. Then there's the Cantor *function* \phi : C --> [0,1] which maps C onto the unit interval. The restriction of \phi to the set C \intersect (0,1) maps that set onto (0,1). So the function f: C \intersect (0,1) --> R defined by f(x) = tan(\pi(\phi(x) - 1/2)) maps the set C \intersect (0,1) onto R. Note that this smears a set of Lebesgue measure 0 across the entire real line. Now, the complement in (0,1) of C \intersect (0,1) is an open set, so it is the disjoint union of countably many disjoint open intervals -- in each of those intervals, you can repeat the above construction (suitablt scaled and translated) to extend the definition of f over each of countably many sets of measure 0, each of which the extended f will map *onto* R. Continue by induction. At the end, you have defined f on a countable collection of disjoint sets of measure zero, each of which it maps onto R. And this collection has the property that every open interval in (0,1) contains one of the sets in the collection. Finish defining f on (0,1) by setting it to 0 wherever it isn't already defined. Then this function maps any nonempty open interval in its domain onto R, which implies that it has the intermediate value property that you required. And that fact *also* implies that f is not continuous at any point of its domain. |> [Note: This isn't homework. I'm just doing some advance summer reading |> in preparation for my start as a mathematics PhD student in |> September.] Good luck with that! -- Ed Hook | Copula eam, se non posit Computer Sciences Corporation | acceptera jocularum. NAS, NASA Ames Research Center | All opinions herein expressed are Internet: hook@nas.nasa.gov | mine alone ============================================================================== From: John Rickard Subject: Re: Nowhere-continuous function with the intermediate value property Date: 05 Jul 2000 19:41:06 +0100 (BST) Newsgroups: sci.math Ed Hook wrote: : Then this function maps any nonempty open interval : in its domain onto R, which implies that it has the : intermediate value property that you required. And : that fact *also* implies that f is not continuous : at any point of its domain. An alternative way to construct such a function, perhaps easier to follow but less mathematically pleasing: Given any x in R, consider the decimal representation of x (for definiteness, not finishing with infinitely many 9s). If this has a tail consisting of an 8 or 9 followed by only 0s, 1s, and exactly one 6, then f(x) is the value of this tail read as a binary number, with 8, 9, and 6 representing "+", "-", and "." respectively. (For example, f(5.123456789116001001001001...) is -11.001001001001... (binary), which is -22/7.) Otherwise, f(x) = 0. -- John Rickard ============================================================================== From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Nowhere-continuous function with the intermediate value property Date: 5 Jul 2000 14:52:23 -0400 Newsgroups: sci.math In article , J. Antonio Ramirez R. wrote: : :Is there such a thing? : :By the intermediate value property, I mean that if f(x)=a and f(y)=b :for some x0 in binary system: x = m + a_1/2^1 + a_2/2^2 + ... with m non-negative integer, and every a_k either 0 or 1. Avoid expansions with a tail of all 1's. Then g(x) = lim sup (a_1 + a_2 + ... + a_n)/n and f(x) = g(|x|). Cheers, ZVK(Slavek) ==============================================================================\ From: ramirez@theory.lcs.mit.edu (J. Antonio Ramirez R.) Subject: Re: Nowhere-continuous function with the intermediate value property Date: 05 Jul 2000 15:04:48 -0400 Newsgroups: sci.math I received a couple of very useful replies from Lars Olsen and Matthew Wiener; thanks! All that is required is a function R->R that is onto on each interval. I received two different constructions by email, but I think it's enough to say that one such function is described in _Counterexamples in analysis_, by Gelbaum and Olmsted. My real motivation for asking the question was solving this problem: given f:[0,1]->R everywhere differentiable, show that f' is somewhere continuous. I was hoping that the intermediate value property of f' would be enough to show this, but the example shows it's not. What follows is Lars Olsen's two-line proof, with some useful references (with his permission). - begin email quote -------------------------------------------------- Date: Wed, 5 Jul 2000 18:47:06 +0000 To: Antonio Ramirez From: Lars Olsen Subject: Re: your sci. math question [...] Here is a two-line proof the Proposition: If f:[0,1]->R everywhere differentiable, then f' is somewhere continuous Proof: The derivative is trivially a Baire 1 function and every Baire function has a dense set of continuity points, hence f' has a dense set of continuity points. Of course, this proof requires that you know what a Baire 1 function is (a simple way to define Baire 1 functions is: a function is called baire 1 if it is the _pointwise_ of a squence of continuous functions), and it requires that you know the fact that every Baire function has a dense set of continuity points. Best regards Lars [...] - end quote ---------------------------------------------------------- - begin email quote -------------------------------------------------- To: Antonio Ramirez From: Lars Olsen Subject: Re: your sci. math question The realm of Baire 1 (and more gennereally Baire n) functions is descriptive set theory. A good reference is Kechris, Classical Desciptive Set Theory, Springer. However, Hewitt & Stromberg, Classical and Abstract Analysis, contains a selfcontained proof of the fact that every Baire 1 function has a dense set of continuity points (it appears as an exercise with appropiate hints). Regards Lars [...] - end quote ---------------------------------------------------------- -- J. Antonio Ramirez R. Remove hybrid silverware from email address. ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: Nowhere-continuous function with the intermediate value property Date: 6 Jul 2000 16:51:00 -0400 Newsgroups: sci.math J. Antonio Ramirez R. [sci.math 05 Jul 2000 12:47:34 -0400] wrote > Is there such a thing? > > By the intermediate value property, I mean that if f(x)=a and > f(y)=b for some x by some z in [x,y]. > > I am guessing it can be constructed using "condensation of > singularities" from the function sin(1/x), in the same manner > that a nowhere-differentiable function can be constructed with > this method from the absolute value function. I can elaborate > on request, but if someone knows what I'm talking about, I would > be grateful to hear whether the idea works. > > [Note: This isn't homework. I'm just doing some advance summer > reading in preparation for my start as a mathematics PhD student > in September.] and then, in a later post, wrote (in part): [sci.math 05 Jul 2000 15:04:48 -0400] > My real motivation for asking the question was solving this > problem: given f:[0,1]->R everywhere differentiable, show that > f' is somewhere continuous. I was hoping that the intermediate > value property of f' would be enough to show this, but the > example shows it's not. The method of condensation of singularities only spreads point-wise (pathological) behavior onto countable sets (which can be dense, of course). Some additional remarks can be found in my sci.math post funky function [Sept. 4, 1999] . More generally, no method will give you a derivative not satisfying the intermediate value property. I see that several posters have already pointed this out, but you may also find the following posts of mine useful: {Includes a proof that every derivative satisfies the intermediate value property.} Why is an integral the antiderivative? [March 18, 2000] {Some remarks about pathological functions having the intermediate value property are given in my comments to the original poster's 2'nd question.} real analysis [Nov. 25, 1999] {Example of a function satisfying the intermediate value property that is not Riemann integrable over any closed subinterval of [0,1].} [MATHEDU] IVProperty [My post is contained in David Epstein's Jan. 18, 2000 post. Epstein's inserted comment about one of my remarks is true, but not relevant since the empty set is a G_delta set. (Also, my comment that the points of continuity form a residual set is only made for the specific function at hand.)] If you're interested in more information about Darboux functions (what functions satisfying the intermediate value property are called), the following books will be helpful: Ralph P. Boas, A PRIMER OF REAL FUNCTIONS, 4'th edition (revised and updated by Harold P. Boas), The Carus Mathematical Monographs #13, The Mathematical Association of America, 1996. A. C. M. Van Rooij and W. H. Schikhof, A SECOND COURSE ON REAL FUNCTIONS, Cambridge University Press, 1982. Andrew M. Bruckner, DIFFERENTIATION OF REAL FUNCTIONS, CRM Monograph Series #5, American Mathematical Society, 1994. [QA 304 .B78 1994] {See chapter 1.} The place to begin a serious study of Darboux functions is the (now a bit dated) survey paper Andrew M. Bruckner and Jack G. Ceder, "Darboux Continuity", Jahresbericht Deutschen Mathem.-Vereinigung 67 (1965), 93-117. As for your follow-up comments about derivatives always having a point of continuity (you mentioned that Lars Olsen has already told you "YES", because derivatives are Baire one functions), you might find some useful information in my fairly lengthy (32 references are given in it) sci.math post HISTORICAL ESSAY ON CONTINUITY OF DERIVATIVES [Jan. 23, 2000] . Dave L. Renfro