Naughty Functions

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Naughty Functions

Postby Tac-Tics » Thu Sep 25, 2008 6:43 pm UTC

I'm thinking about writing something about the role of functions in mathematics. One part of it will be dedicated to "weird" functions. I'm asking xkcd to help me come up with some good examples. That is, I'm looking for any function that is misbehaved or atypical or just plain cool that most people don't think of as a function. Nonelementary functions. Functions which are continuous but nowhere smooth. Functions that have any counter-intuitive properties. Real functions are best, but easy-to-explain functions on other domains work too.

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Re: Naughty Functions

Postby Token » Thu Sep 25, 2008 7:13 pm UTC

The Weierstrass function is your standard example.
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Re: Naughty Functions

Postby Torn Apart By Dingos » Thu Sep 25, 2008 7:14 pm UTC

The indicator function function of the rationals. Nowhere continuous.
The popcorn function, continuous only on the irrationals.
The Busy Beaver function.
Bijections from N to N^2. One that is given by a formula is f(n,m)=(n+m)*(n+m+1)/2+n (it's enumerates the elements on the diagonals). I haven't seen any neat bijections from R to R^2.
Spacefilling curves.
The devil's staircase: a continuous, monotonous function with zero derivative almost everywhere, but it increases from 0 to 1 on [0,1].
A well-ordering on the reals.
[imath]f(x)=\begin{cases}\exp(-1/x^2), & x\geq 0\\0,&x\leq 0\end{cases}[/imath] is infinitely differentiable but not analytic.
Additive (i.e. f(x+y)=f(x)+f(y)) functions that aren't linear.
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Re: Naughty Functions

Postby SunAvatar » Thu Sep 25, 2008 7:15 pm UTC

I recommend, when you talk about "weird functions," that you point out the not-quite-intuitive fact that, in the sense of cardinality, most functions are poorly behaved.

Here are some obvious starters, all with domain and codomain [imath]\mathbb{R}[/imath]:
  • The indicator function [imath]1_\mathbb{Q}[/imath] for [imath]\mathbb{Q}[/imath] in [imath]\mathbb{R}[/imath] (1 if rational, 0 if irrational) is defined everywhere but continuous nowhere.
  • The Weierstrass function (http://en.wikipedia.org/wiki/Weierstrass_function) is continuous everywhere but differentiable nowhere.
  • The function [math]f(x) = \begin{cases}\ \ x^2 & x \in \mathbb{Q} \\ \ \ 0 & x \not\in \mathbb{Q}\end{cases}[/math] is continuous only at 0---and it is differentiable at 0.
  • Thomae's function (http://en.wikipedia.org/wiki/Popcorn_function) is continuous at all irrational numbers and discontinuous at all rational numbers.
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Re: Naughty Functions

Postby Hix » Thu Sep 25, 2008 7:17 pm UTC

There's a book called "Counterexamples in Analysis" that's just full of this kind of stuff; you may find it useful if you can find a copy.

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Re: Naughty Functions

Postby Tac-Tics » Thu Sep 25, 2008 7:21 pm UTC

Hix wrote:There's a book called "Counterexamples in Analysis" that's just full of this kind of stuff; you may find it useful if you can find a copy.


I thought of that book when I was writing this. However, why go out of my way to browse through it in the bookstore if I can bum the more familiar examples off of bored xkcd forumgoers ;-)

everyone else wrote:(stuff about continuity)


Thank you for your help. It's good to have the names of those functions so I can look into them. Can anyone think of any more weird functions. Ones that don't necessarily have to do with continuity. I'm looking for variety =-)

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Re: Naughty Functions

Postby Ended » Thu Sep 25, 2008 7:28 pm UTC

SunAvatar wrote:The indicator function 1Q for Q in R (1 if rational, 0 if irrational)

[math]1Q(x) = lim_{n \to \infty} lim_{m \to \infty} cos^{m}(2n \pi x)[/math]

Do your functions have to be constructive? I remember an exercise which proved non-constructively that there exists a smooth function f: [0,1] -> R whose Taylor series around any point t diverges in any open interval containing t. Crazy stuff.
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Re: Naughty Functions

Postby Yakk » Thu Sep 25, 2008 7:51 pm UTC

The Banach–Tarski paradox has a few functions that are interesting, and require choice to exist.
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Re: Naughty Functions

Postby SunAvatar » Thu Sep 25, 2008 8:05 pm UTC

Most of the ways that functions in general can be 'nice' are rather more esoteric than continuity or differentiability---that is, even though they're nice things to have, a naïve observer wouldn't necessarily expect them by default. No one really expects all functions to be one-one or onto.

You could point out that the derivative is itself a function, from the set of differentiable real-valued functions to the set of real-valued functions.

You could consider two ways of looking at a binary operation like addition: as a function from [imath]\mathbb{R} \times \mathbb{R}[/imath] to [imath]\mathbb{R}[/imath], as is usually done, or in its curried form as a function from [imath]\mathbb{R}[/imath] to the set of functions from [imath]\mathbb{R}[/imath] to [imath]\mathbb{R}[/imath]. (That is, '+' is a function that, applied to 3, gives '3+', a function that, applied to 4, gives 7.) In either case, a binary operation is a function.

You could give a cardinality argument that, within any given fixed language, 'most' real-valued (or indeed even integer-valued) functions have infinite Komolgorov complexity, and so can never actually be written down specifically. (This helps clarify what is imprecise about the informal "A function is a rule...." statement.)

You could point out that, for any set [imath]S[/imath], there is exactly one function from the empty set to [imath]S[/imath]. (Depending on how one formally defines a function, this function may itself be the empty set.)
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Re: Naughty Functions

Postby skeptical scientist » Thu Sep 25, 2008 8:15 pm UTC

Some easy examples from calculus:
sin(1/x) - has an essential discontinuity at 0.
xsin(1/x) - is continuous (if you remove the removable discontinuity at 0), but not differentiable at 0.
x2sin(1/x) - is everywhere differentiable, but has a non-continuous derivative.

One example that hasn't yet been mentioned is a function which is differentiable but whose derivative is not Riemann integrable. A construction can be found in, e.g., Abbott, Understanding Analysis, section 7.6. The idea behind his construction is this:

Let C be a fat Cantor set. The complement of C is a union of disjoint open intervals. We define f by setting f=0 on C, and for each of the open intervals that makes up the complement of C, paste in something that is differentiable but has a scaled copy of x2sin(1/x) at each end. The resulting function is everywhere differentiable, but the derivative has discontinuities at every point of C. Since the derivative is discontinuous on a set of positive measure, it is not Riemann integrable (by Lebesgue's criterion for Riemann integrability).
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Re: Naughty Functions

Postby Ended » Thu Sep 25, 2008 8:32 pm UTC

skeptical scientist wrote:sin(1/x) - has an essential discontinuity at 0.

That reminds me - Picard's great theorem is seriously weird and cool. It says that if an analytic function f(z) has an essential singularity at z = w, then f takes all possible complex values (with possibly one exception) infinitely often, in any open set containing w.

You can apply it to something simple like sin(1/z) or e1/z, with w = 0.
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Re: Naughty Functions

Postby Token » Thu Sep 25, 2008 8:57 pm UTC

Ended wrote:
skeptical scientist wrote:sin(1/x) - has an essential discontinuity at 0.

That reminds me - Picard's great theorem is seriously weird and cool. It says that if an analytic a meromorphic function f(z) has an essential singularity at z = w, then f takes all possible complex values (with possibly one exception) infinitely often, in any open set containing w.

You can apply it to something simple like sin(1/z) or e1/z, with w = 0.

Fix'd. Analytic functions can't technically have singularities.
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Re: Naughty Functions

Postby troyp » Thu Sep 25, 2008 9:39 pm UTC

These are interesting functions: I've never really looked into "weird" functions, although I've wondered about them.

I have a question about the Weierstrauss function: does its graph have infinite length between any 2 x-ordinates? If so, is this true for any nowhere-differentiable functions?
(I'm not sure how you'd even deal with the length of a function like that - I'm guessing you could take a line of any length and bend it so it's a smoothed version of the graph, and that the graph therefore has to have infinite length)

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Re: Naughty Functions

Postby antonfire » Thu Sep 25, 2008 9:49 pm UTC

Token wrote:
Ended wrote:
skeptical scientist wrote:sin(1/x) - has an essential discontinuity at 0.

That reminds me - Picard's great theorem is seriously weird and cool. It says that if an analytic a meromorphic function f(z) has an essential singularity at z = w, then f takes all possible complex values (with possibly one exception) infinitely often, in any open set containing w.

You can apply it to something simple like sin(1/z) or e1/z, with w = 0.

Fix'd. Analytic functions can't technically have singularities.
Huh? Meromorphic functions aren't allowed to have essential singularities. Sure, "analytic" is sometimes used to mean "analytic on C" (the proper term for this is "entire", I think), but here it's clear from context that that's not what skeptical means.
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Re: Naughty Functions

Postby Token » Thu Sep 25, 2008 10:15 pm UTC

Bah. I forgot the condition that meromorphic functions can have at worst poles. I shall stop being so pedantic about areas of mathematics I tended to sleep through. Not my fault if Complex Analysis is a hugely boring subject. "Analytic on D(w,r) \ {w} for some r > 0" then, if you insist. And it wasn't skeptical.
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Re: Naughty Functions

Postby jestingrabbit » Thu Sep 25, 2008 10:38 pm UTC

troyp wrote:I have a question about the Weierstrauss function: does its graph have infinite length between any 2 x-ordinates? If so, is this true for any nowhere-differentiable functions?


The length of a curve is defined in terms of the derivative, so I would say that the lenght of the weierstrass function is undefined, but you could probably make an argument for it being infinite.
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Re: Naughty Functions

Postby Kalak_z » Thu Sep 25, 2008 10:52 pm UTC

How about a function that is unbounded in any open interval, but has a finite integral over all of R.

Let [imath]\{q_n\}[/imath] be an enumeration of the rational numbers. Let [math]g(x) = \left\{ \begin{array}{cc} \sqrt x -1 & \;\;\; 0<x \leq 1 \\ 0 & \textrm{else}\end{array}\right. .[/math] Finally, let [imath]f(x) = \sum_n 2^{-n}g(x-q_n)[/imath]. The function [imath]f[/imath] has a pole (from the right) at each rational number, yet [math]\int_{\mathbb R} f = \sum_n 2^{-n} \int_0^1 g = \sum_n 2^{-n} 1 =1.[/math]

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Re: Naughty Functions

Postby Hefty One » Thu Sep 25, 2008 11:07 pm UTC

Ackermann's Function

Not really fun unless you try to evaluate it using Graham's Number

Infinity doesn't stand a chance...

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Re: Naughty Functions

Postby Office_Shredder » Thu Sep 25, 2008 11:31 pm UTC

Counterexamples in Analysis is on that google book thing (google the title and it'll be at the top)

One cool example if a function that's finite everywhere but everywhere unbounded

f(x) = 0 if x is irrational, m if x is rational and of the form m/n when reduced
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Re: Naughty Functions

Postby z4lis » Thu Sep 25, 2008 11:40 pm UTC

Most awesome function ever:
[math]f(x) =
\begin{cases}
0, & \mbox{if }x \notin \mathbb{Q}\\
1/q & \mbox{ if }x= \frac{p}{q} \in \mathbb{Q}
\end{cases}[/math]

It's awesome because it's continuous at every irrational in (0,1), but discontinuous at every rational in that set.
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Re: Naughty Functions

Postby jestingrabbit » Fri Sep 26, 2008 12:14 am UTC

ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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Re: Naughty Functions

Postby Ended » Fri Sep 26, 2008 8:15 am UTC

jestingrabbit wrote:http://en.wikipedia.org/wiki/Minkowski's_question_mark_function

Hehe, all the question marks in the formulae make it look like they're just guessing things.
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Re: Naughty Functions

Postby lgonick » Fri Sep 26, 2008 2:04 pm UTC

How about a continuous function f with finite integral as x --> infinity but f(x) doesn't --> 0 as x -> infinity? An example would be f(x) = 0 everywhere except for intervals of length (1/2)^n centered on n. On those intervals, f rises to 1 and returns to 0 in series of ever-narrower triangular spikes. I'm too lazy to write down the formula.

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Re: Naughty Functions

Postby lgonick » Fri Sep 26, 2008 3:40 pm UTC

Based on my previous post, how about this (easy?) exercise:

I described a function identically = 0 on an interval that then rises continuously at the interval's end. Now let's try smoothing it out at the corners. So... imagine any old function f and a real number a such that

f(x) = 0 for x ≤ a
f(x) > 0 for x > a

Show that you can construct such a function with an arbitrary number of derivates at a. That is, given n, I can make an n-times differentiable function f with those properties.

But no such function can be infinitely differentiable.

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Re: Naughty Functions

Postby jestingrabbit » Fri Sep 26, 2008 5:05 pm UTC

lgonick wrote:Based on my previous post, how about this (easy?) exercise:

I described a function identically = 0 on an interval that then rises continuously at the interval's end. Now let's try smoothing it out at the corners. So... imagine any old function f and a real number a such that

f(x) = 0 for x ≤ a
f(x) > 0 for x > a

Show that you can construct such a function with an arbitrary number of derivates at a. That is, given n, I can make an n-times differentiable function f with those properties.

But no such function can be infinitely differentiable.


Bull.

f(x) = 0 for x<= a
= exp(-1/(x-a)2)

The guts of f were described earlier on this very page.
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Re: Naughty Functions

Postby stephentyrone » Fri Sep 26, 2008 7:37 pm UTC

Personal favorite:

[math]f(x,y) = \left\{\begin{array}{ll}\frac{x^2y}{x^4 + y^2}&(x,y)\neq(0,0)\\0&(x,y)=(0,0)\end{array}\right.[/math]

Continuous along every straight line path, but discontinuous at (0,0); the limit as [imath](x,y)\rightarrow(0,0)[/imath] along the path [imath]y = \pm x^2[/imath] is [imath]\pm\frac12[/imath].
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Re: Naughty Functions

Postby Hefty One » Fri Sep 26, 2008 11:19 pm UTC

Not really naughty but more anecdotal, but not something I see every day.

[math]y= x^\frac {\left |x \right |}{x} + \frac {1}{x}[/math]

Function with all three asymptotes.

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Re: Naughty Functions

Postby lgonick » Sat Sep 27, 2008 1:55 am UTC

Right you are, and wrong I was!

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Re: Naughty Functions

Postby skeptical scientist » Sat Sep 27, 2008 2:16 am UTC

lgonick wrote:How about a continuous function f with finite integral as x --> infinity but f(x) doesn't --> 0 as x -> infinity? An example would be f(x) = 0 everywhere except for intervals of length (1/2)^n centered on n. On those intervals, f rises to 1 and returns to 0 in series of ever-narrower triangular spikes. I'm too lazy to write down the formula.

The function sin(x2) works. Use the alternating series test.
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Re: Naughty Functions

Postby kgrizzly » Sat Sep 27, 2008 8:03 am UTC

How about generating functions? Which, I suppose, are not functions in the usual sense. Or perhaps it's better to say is that the functions themselves may not be weird, it's how they are used that seems weird when you first encounter them. The best reference on these, for beginners anyway, is Herb Wilf's "generatingfunctionology" (yes, it's all one word) which is freely available as a PDF file on his website.

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Re: Naughty Functions

Postby Charlie! » Sat Sep 27, 2008 8:15 pm UTC

skeptical scientist wrote:
lgonick wrote:How about a continuous function f with finite integral as x --> infinity but f(x) doesn't --> 0 as x -> infinity? An example would be f(x) = 0 everywhere except for intervals of length (1/2)^n centered on n. On those intervals, f rises to 1 and returns to 0 in series of ever-narrower triangular spikes. I'm too lazy to write down the formula.

The function sin(x2) works. Use the alternating series test.

I thought the absolute value of alternating series had to be constantly decreasing for that to work?
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Re: Naughty Functions

Postby Hix » Sat Sep 27, 2008 8:30 pm UTC

Charlie! wrote:
skeptical scientist wrote:
lgonick wrote:How about a continuous function f with finite integral as x --> infinity but f(x) doesn't --> 0 as x -> infinity? An example would be f(x) = 0 everywhere except for intervals of length (1/2)^n centered on n. On those intervals, f rises to 1 and returns to 0 in series of ever-narrower triangular spikes. I'm too lazy to write down the formula.

The function sin(x2) works. Use the alternating series test.

I thought the absolute value of alternating series had to be constantly decreasing for that to work?

Well, he didn't spell it out, but the "alternating series" that skeptical is talking about is probably:
sin(x2) is positive from 0 to sqrt(pi), and encloses a certain area;
sin(x2) is negative from sqrt(pi) to sqrt(2pi), and clearly encloses a smaller area;
sin(x2) is positive from sqrt(2pi) to sqrt(3pi), and encloses an area that's even smaller than that;
...

so the sum of all the (signed) areas from 0 to infinity will converge.

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Re: Naughty Functions

Postby skeptical scientist » Sun Sep 28, 2008 12:02 am UTC

Hix wrote:
Charlie! wrote:
skeptical scientist wrote:
lgonick wrote:How about a continuous function f with finite integral as x --> infinity but f(x) doesn't --> 0 as x -> infinity? An example would be f(x) = 0 everywhere except for intervals of length (1/2)^n centered on n. On those intervals, f rises to 1 and returns to 0 in series of ever-narrower triangular spikes. I'm too lazy to write down the formula.

The function sin(x2) works. Use the alternating series test.

I thought the absolute value of alternating series had to be constantly decreasing for that to work?

Well, he didn't spell it out, but the "alternating series" that skeptical is talking about is probably:
sin(x2) is positive from 0 to sqrt(pi), and encloses a certain area;
sin(x2) is negative from sqrt(pi) to sqrt(2pi), and clearly encloses a smaller area;
sin(x2) is positive from sqrt(2pi) to sqrt(3pi), and encloses an area that's even smaller than that;
...

so the sum of all the (signed) areas from 0 to infinity will converge.

Right. That tells you that if you consider the sequence [imath]\int_0^\sqrt{n\pi} \sin{x^2} \, \text{d}x[/imath], it converges to some limit L. Furthermore, since the integral can't change very much between [imath]\sqrt{n\pi}[/imath] and [imath]\sqrt{(n+1)\pi}[/imath] (since the function is bounded by 1, and [imath]\lim_n (\sqrt{(n+1)\pi}-\sqrt{n\pi})=0[/imath]), this tells you that the indefinite integral converges.

Also, you can use methods of complex analysis to prove the limit is [imath]\sqrt{\frac{\pi}{8}}[/imath] - see Wikipedia.
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Re: Naughty Functions

Postby lgonick » Sun Sep 28, 2008 2:12 pm UTC

OK, then, a non-negative continuous function that doesn't -> 0 but has finite integral as x -> infinity.

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Re: Naughty Functions

Postby MartianInvader » Sun Sep 28, 2008 3:32 pm UTC

I can't think of a way to define it nicely in one equation, but I'll describe such a function for you:

Let f(x) be zero for 0 <= x <= 1. Between x = 1 and x = 2, have it jump up and then down (in two straight lines if you don't mind it being nondifferentiable, or in a bump if you want it smooth), so that f(1.5) = 1. Now, have it do the exact same thing between x = 2 and x = 2.5, except squished by half in the x direction so that it fits in an interval half as big. (Don't squish it in the y direction, so f(2.25) = 1.) Have f(x) = 0 for 2.5 <= x <= 3. Have it jump up to 1 and back again between 3 and 3.25, and then be zero until x = 4.

Continue in this way, with a jump up to f(x) = 1 starting at each integer, but each such bump being half as wide as the bump before, and then f(x) is zero until the next integer. Each bump has an integral equal to half the previous bump, so the integral from zero to infinity of this function will be twice that of your original bump. And since the function keeps jumping up to the value of 1, it does not approach zero as x -> infinity.
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Re: Naughty Functions

Postby Cosmologicon » Sun Sep 28, 2008 3:55 pm UTC

Here's an explicit description using the smooth bump idea:

[math]\sum_{n=1}^{\infty}\exp{\left(-{n^4(x-n)^2}\right)}[/math]

which is greater than 1 on all positive integers. If I did this right, the integral goes like the sum of 1/n^2.

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Re: Naughty Functions

Postby skeptical scientist » Sun Sep 28, 2008 7:39 pm UTC

I thought sin x2 was a nice example because the formula is so simple. What's so great about positive functions anyways? :P
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Re: Naughty Functions

Postby Diadem » Sun Sep 28, 2008 10:34 pm UTC

Cosmologicon wrote:Here's an explicit description using the smooth bump idea:

[math]\sum_{n=1}^{\infty}\exp{\left(-{n^4(x-n)^2}\right)}[/math]

which is greater than 1 on all positive integers. If I did this right, the integral goes like the sum of 1/n^2.


Awesome! Not only is that function positive and does it have a a finite integral without going to zero, it's even strictly positive and differentiable.

I was actually wondering if such a function was possible. I guess you answered my question!
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samspotting
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Re: Naughty Functions

Postby samspotting » Mon Sep 29, 2008 2:50 pm UTC

There was a function that my prof showed that was not well behaved anywhere and it was a summation. does anyone know an example?

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Yakk
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Re: Naughty Functions

Postby Yakk » Mon Sep 29, 2008 5:48 pm UTC

samspotting wrote:There was a function that my prof showed that was not well behaved anywhere and it was a summation. does anyone know an example?

In what way was it not well behaved?
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Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.


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