Question

Is there a function f which is discontinuous at every point, and which has only removable discontinuities? (Extra credits if you can prove your answer)

Answer 1

The book rated this question 'Extra hard' and suggested that you will need some knowledge of infinite sequences in order to do the 'proof' part, but knowing limits would be enough to give a simple answer.

Answer 2

The only way I can think of doing that would be to define the function at every point, and cause it to be discontinuous.

Alternatively, you could cheat:
\[\Gamma(x)*i\]
is a continuous function, but only in the complex plane. Causing it to be discontinuous (and undefined) in the reals... but you could fill these gaps by applying /i to all the y values.

Answer 3

my 2cents worth - what about:\[f(x)=\frac{1}{x(x^2-1)(x^2-4)...(x^n-n^2)...}\]this is discontinuous from -infinity to infinity (including zero). but only at integer values of x. is this what you want?

Answer 4

JamesJ to the rescue!

Answer 5

I'm pretty sure the answer is no.

An example of such a function that is discontinuous everywhere is the following:
\[ f(x) = \left\{\begin{matrix}1 & x \in Q \\ 0 & x \in R - Q \end{matrix}\right. \]

Answer 6

or rather, that we need some additional hypothesis

Answer 7

or shaper hypothesis such as the number of points where the function is discontinuous are countable.

Answer 8

Once that Spivak Answer book arrives I will know the true answer.

The infinite sequence section 400 pages later says something about it not being possible for f to be discontinuous everywhere

Answer 9

what about the indicator function? http://en.wikipedia.org/wiki/Indicator_function

Answer 10

This is in Spivak? One sec

Answer 11

Yes, my intuition was right.

If a function f has only removable discontinuities, then f is continuous except at a countable set of points.

Hence if a function f is discontinuous everywhere, it cannot have only removable discontinuities.

Answer 12

Changing topic, I wish I'd enrolled in the MIT ML course. How's that one going for you?

Answer 13

I like the AI course a lot, but the ML course looks like the applied version of that and I would appreciate the discipline of having to actually code the algorithms.

Answer 14

It's working so great that I finally understand virtually everything in this page: http://david-hu.com/2011/11/02/how-khan-academy-is-using-machine-learning-to-assess-student-mastery.html

Answer 15

but the ML class uses GNU Octave (its like a free version of MATLAB) and not Python.

Answer 16

Even so, having to write a program that using the algorithm I think is essential for really learning the material; I worry that in six months I will have forgotten most of it.

Answer 17

It's analogous to telling someone: given the equation ax^2 + bx + c = 0, where a, b, c are constants and a is not zero then x = -b +- .....

...and then not getting them to actually solve any quadratic equations.

Answer 18

I worry the same thing about the AI class, which has no mandatory programming assignments although the ones assigned to the actual Stanford class are available on the internet.

Answer 19

In other words, there's a huge difference between understanding something at some superficial level of theory, and actually using it, which nearly always deepens the understanding of that theory.

Answer 20

They'd originally thought of giving programming assignments, but when enrollment went over 20,000--- or so the story has it---- that they were worried it wouldn't be pragmatic for them.

Answer 21

:( would have been nice to have some programming assignments.

Answer 22

Agreed. Well, at least you have some in the ML course, so good for you.

Answer 23

The DB class programming assignments are quite non-trivial :(

Answer 24

ha. Well, you signed up!

Answer 25

The subject of discontinuities seems interesting so I looked around to see I could find anything that could explain the concept to me in more detail. Unfortunately it all looks too complex and beyond my grasp.
However, I did come across this article that may (or may not) be of some interest to you as it seems to be related to your question: http://math.stackexchange.com/questions/3777/is-there-a-function-with-a-removable-discontinuity-at-every-point

Answer 26

I must say until 10 days ago, I didn't even understand the Machine Learning was a major topic in AI. Had I known that four weeks ago, I would have signed up for the ML course. Live and learn.

Answer 27

jamesj and agdgdgdgwngo - if you guys need any help in programming that I am more than willing to help - I have over 30 years of experience in the field and use C++, Java, Python.

Answer 28

I need help understanding basic data structures such as linked lists, binary trees, and hash tables.

Answer 29

Have you looked at AI assignment 4 yet? I thought assignment 3 was really too easy. That might be a bastard of a thing to say, but assignment 3 was not at university level at all.

Answer 30

(@asnaneer, thanks.)

Answer 31

Assignment 3 was cake, and I think assignment 4 will be cake too, at least for you :) ; I am averaging 60% in the class quizzes :(.

Answer 32

I'm still planning to stick and work hard on the AI class, for this: http://www.aaai.org/Membership/ai-course-offer.php

Answer 33

Don't worry about the class quizzes; the whole point of those is to make you think through the material. I made quite a few errors in unit 5 quizzes at first, mainly because it wasn't intuitively obviously how to do some of those probability calculations until you've seen one. But after you've seen it, it's pretty straight forward.

Answer 34

So for example unit 6 was back on familiar mathematical ground for me and that was straightforward.

Answer 35

I was joking with a good friend of mine that I haven't actually used the formula for regression coefficients for over a decade; it's something I do all the time, but I always have it done for me computationally or by someone working for me. It was a bit like going back to high school where we had to calculate those things without using stat functions on calculators.

Answer 36

Right. The AI class instructors have a somewhat different teaching style compared to Andrew Ng's ML class; they explain the 'basics' and then follow up with a relatively challenging quiz :) while Ng anticipates the hard stuff and explains it first to build your intuition before the quizzes.

Answer 37

So, for the material that was new, I go back through the quizzes again the next day and make sure I've got it down.

Answer 38

I have to tell you, if nothing else, the AI course has turned into a Bayesian calculating machine. I won't forget that skill in six months.

Answer 39

james, that function is discontinuous everywhere

Answer 40

the x =1 , x = 0 one

Answer 41

@sarah: yes, I know.

Answer 42

Scroll up to find our final conclusion.

Answer 43

The following:

If a function f has only removable discontinuities, then f is continuous except at a countable set of points.

Hence if a function f is discontinuous everywhere, it cannot have only removable discontinuities.

Answer 44

hmmm

Answer 45

We haven't give here, but you'll find it as a relatively sophisticated exercise--and broken down in steps--in Spivak, chapter 22.

Answer 46

if f has removable discontinuues... why cant it be non countable discontinuous

Answer 47

so you only proved one statement, and other is the contrapositive?

Answer 48

it is sufficient to prove the first part,
If a function f has only removable discontinuities, then f is continuous except at a countable set of points.

Answer 49

I haven't proved either here, but yes, they are contrapositives of each other.

But I believe it.

Answer 50

what are you guys talking about bayesian calculator and regression coefficients, what is that?

Answer 51

how do you learn about bayesian calculator

Answer 52

We're both taking an on-line course offered by Stanford University on Artificial Intelligence. Bayesian probability is ground-level skill for thinking through a wide range of AI topics, and regression comes up in some context also. That's all.

You can find it here: ai-class.com

Answer 53

is it a free course?

Answer 54

Anyway, @agdgd, good to talk to you. See you soon.

Answer 55

Yes. Enrollment however is now closed.

Answer 56

oh damn

Answer 57

when does it start up again?

Answer 58

not yet announced.

Answer 59

ok the question can be answered more concisely as
is there exist a function which has a removable discontinuity at every point

Answer 60

so clearly it is discontinuous at every point

Answer 61

i would say, no , because to have removable discontinuity you need to talk about the limit around a point , ie, the neighborhood. since there is no neighborhood , therefore there is no limit around a point .

Answer 62

"since there is no neighborhood" doesn't make sense, however I think your intuition is still pointing in the right direction.

Answer 63

anyway, I am going now. Bye

Answer 64

there is no neighborhood around a point

Answer 65

, oh wait, nevermind

Answer 66

sorry i was thinking there were undefined points for some reason

Answer 67

so assuming that it is discontinuous , then the limit does not exist at any point. and for removable discontinuiity you need at least that the limit exists