Continuity of a function.

Most of us, would have studied(likely in high school) about the idea of functions being continuous.

As the wikipedia section states, we end up with 3 conditions for a function over an interval [a,b].

  • The function should be defined at a constant value c
  • The limit has to exist.
  • The value of the limit must equal to c.


Now this is a perfectly useful notion for most of the functions we encounter in high school. But there are functions that would satisfy these three conditions, but still won’t be helpful for us to move forward. And I just discovered while reading up for my self-educating on statistics. One moment there I was trying to understand what’s the beta distribution, or for that matter what sense does it make to talk about a probability density function, I mean understand what’s probability, but how can it have density and that sort of thing.. I lose focus a few seconds, and find myself tumbling down a click-hole to find a curiouser idea about 3 levels/types(ordered by strictness) of continuity of a function namely


The last one being what we studied and what I described above.
Now let’s get Climb up one more step on the ladder of abstraction and see what’s uniform continuity.
Ah we have five more types of continuous there namely

Ok I won’t act vogonic and try to understand or explain all of those . I just put them out there to tease feel free to click your way in.

To quote first line from the uniform continuous wiki:

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves. For instance, any isometry (distance-preserving map) between metric spaces is uniformly continuous.

So what does this mean and how does it differ from the ordinary continuity? Well they say it up there that the maximum distance between f(x) and f(y) cannot depend on x and y themselves. i.e: the dist.function: df(f(x), f(y)) has neither x or y in it’s expression/input/right hand side.

The more formal definition can be quoted like this:

Given metric spaces (X, d1) and (Y, d2), a function f : X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d1(x, y) < δ, we have that d2(f(x), f(y)) < ε.

Now why would this be relevant or useful and why is it higher/stricter than ordinary continuity. Well note that it doesn’t say anything about an interval. The notion of ordinary continuity is always defined on an interval in the input space and clearly is confined to that. i.e: it is a property that is local to the given interval in input/domain space and may or may not apply on other different intervals.

On the other hand if you can say the function is uniformly continuous you’re effectively saying the function is continuous at all intervals.

Now how do we find a more general definition(i.e: absolute continuity?) Well look at the 3 conditions we defined at the start of this blog post. The first two can be collapsed to say the function must be differentiable over the given interval [a,b]. The third is the distance/measure concept we used in uniform continuous definition to remove the bounds on the interval and say everywhere. So obviously for the absolute continuous definition we do the next step and say the function must be uniformly continuous and differentiable everywhere.(aka uniformly differentiable).

Ok all of this is great, except where the hell is this useful. I mean are there function that belong to different continuous classes, so that these definitions/properties and theorems built on these can be used to differentiate and reason about functions. Turns out there are . I’ll start with something i glimpsed on my way down the click-hole, the cantor distribution. . It’s the exception that causes as to create a new class of continuous. It’s neither discrete nor continuous.

It’s distribution therefore has no point mass or probability mass function or probability density function.* Therefore throws a lot of the reasoning/theorem systems for a loop.

For the other example i.e: something ordinarily continuous but not uniformly continuous see here. . It’s a proof by contradiction approach.

* — Ok, I confess, the last point about point mass and probability mass/density function still escapes me. I’ll revisit that later, perhaps this time with the help of that excellent norvig’s ipython notebook on probability.

September 6 – OMG! GMO! — Little facts about science

Today’s factismal: We’ve been using genetically-modified organisms to save lives for 38 years. If you keep up with the news, you are aware that there is a lot of arguing going on over the use of genetically-modified organisms, also known as GMOs to the acronym-lovers out there. On the one hand, there are those who […]

via September 6 – OMG! GMO! — Little facts about science

Indian startups advice — 3.0

As I tend to be brash and stupid and offer advice/criticisms about running startups before(here and here and here.), I’ll do it again.

I shall dispense this advice now.

1. Treat your early employees more like partners than wage slave.*

2. This follows from the previous one. After every hire(and fire) re-consider your selection process.

3. Remember the Charlie Munger advice on trust here (or quoted below)**.

4. The best problem solvers, prefer to focus on solving the problem(s) and go right on to the next problem. They would much rather leave the performance reviews, raises (promised at the time of joining) etc.. to others. So if you do promise any review and raise based on that, follow through, Don’t delay with “we’ll do this in a formal setting in two weeks” dodge and then fail to follow through. You won’t build the best possible team with that approach.

5. Find the product/market fit..(Meh. I’m not qualified to say much about this without hands-on finding one).

5. Build a monopoly niche. Don’t compete on price, use your skills and knowledge to build a big manic monopoly, that would be the biggest barrier of entry to any competitors.
By the way the last two are just me re-gurgitating what I think makes sense from what I have read around. Only currently experimenting with implementing them.

** – “The highest form that civilization can reach is a seamless web of deserved trust — not much procedure, just totally reliable people correctly trusting one another. … In your own life what you want is a seamless web of deserved trust. And if your proposed marriage contract has forty-seven pages, I suggest you not enter.”
Source: Wesco Financial annual meeting, 2008 (quoted in Stanford Business School paper)

* — Note how I didn’t say anything about politeness or good salary or on time salary etc. That’s because all of those can be wrong ones to emphasize. My whole reason for this point is that they should have skin-in-the-game. Everything else can be worked around. Just don’t do this.

Few other links:

Bayesians vs Frequentists(aka sampling theorists)

This is a long standing debate/argument and like most polarized arguments, both sides have some valid and good reasons for their stand. (There goes the punchline/ TLDR). I’ll try to go a few levels deeper and try to explain the reasons why I think this is kind of a fake argument. (Disclaimer: am just a math enthusiast, and a (willing-to-speculate) novice. Do your own research, if this post helps as a starting point for that, I’d have done my job.)

  • As EY writes in this post about how bayes theorem is  a law that’s more general and should be observed over whatever frequentist tools we have developed?
  • If you read the original post carefully, he doesn’t mention the original/underlying distribution, guesses about it or confidence interval(see calibration game)
  •  He points to a chapter(in the addendum) here.
  • Most of the post otherwise is about using tools vs using a general theory and how the general theory is more powerful and saves a lot of time
  •  My first reaction to the post was but obviously there’s a reason those two cases should be treated different. They both have the same number of samples, but different ways of taking the samples. One sampling method(one who does sample till 60% success) is a biased way of gathering data .
  • As a different blog and some comments point out, if we’re dealing with robots(deterministic algorithmic data-collector) that precisely take data in a rigourous deterministic algorithmic manner the bayesian priors are the same.
  • However in real life, it’s going to be humans, who’ll have a lot more decisions to make about considering a data point or not. (Like for example, what stage of the patient should be when he’s considered a candidate for the experimental drug)
  • The point I however am going to make or am interested in making is related to known-unknowns vs unknown-unknowns debate.
  • My point being even if you have a robot collecting the data, if the underlying nature of the distribution is unknown-unknown(or for that matter depends on a unknown-unknown factor, say location, as some diseases are more widespread in some areas)  mean that they can gather same results, even if they were seeing different local distributions.
  • A contiguous point is that determining the right sample size is a harder problem in a lot of cases to be confident about the representativeness of the sample.
  • To be fair, EY is not ignorant of this problem described above. He even refers to it a bit in his 0 and 1 are not probabilities post here. So the original post might have over-simplified for the sake of rhetoric or simply because he hadn’t read The Red queen.
  • The Red queen details about a bunch of evolutionary theories eventually arguing that the constant race between parasite and host immune system is why we have sex  as a reproductive mechanism and we have two genders/sexes.
  • The medicine/biology example is a lot more complex system than it seems so this mistake is easier to make.
  • Yes in all of the cases above, the bayesian method (which is simpler to use and understand) will work, if the factors(priors) are known before doing the analysis.
  • But my point is that we don’t know all the factors(priors) and may not even be able to list all of them, let alone screen, and find the prior probability of each of them.

With that I’ll wind this post down. But leave you with a couple more posts I found around the topic, that seem to dig into more detail. (here and here)


P.S: Here’s a funny Chuck Norris style facts about Eliezer Yudkowsky.(Which I happened upon when trying to find the original post and was not aware of before composing the post in my head.) And here’s an xkcd comic about frequentists vs bayesians.

UPDATE-1(5-6 hrs after original post conception): I realized my disclaimer doesn’t really inform the bayesian prior to judge my post. So here’s my history/whatever with statistics. I’ve had trouble understanding the logic/reasoning/proof behind standard (frequentist?) statistical tests, and was never a fan of rote just doing the steps. So am still trying to understand the logic behind those tests, but today if I were to bet I’d rather bet on results from the bayesian method than from any conventional methods**.

UPDATE-2(5-6 hrs after original post conception):  A good example might be the counter example. i.e: given the same data(aka in this case frequency of a distribution, nothing else really, i.e: mean, variance, kurtosis or skewness) show that bayesian method gives different results based on how it(data) was collected and frequentist doesn’t. I’m not sure it’s possible though given the number of methods frequentist/standard methods use.

UPDATE-3 (a few weeks after original writing): Here’s another post about the difference in approaches between the two.

UPDATE-4 (A month or so after): I came across this post with mentions more than two buckets, but obviously they are not all disjoint sets(buckets).

UPDATE-5(Further a couple of months after): There’s a slightly different approach to splitting the two cultures from a different perspective here.
** — I might tweak the amount I’d bet based on the results from it .

In fact, any time anybody offers you anything with a big commission and a 200-page prospectus, don’t buy it. Occasionally, you’ll be wrong if you adopt “Munger’s Rule”. However, over a lifetime, you’ll be a long way ahead—and you will miss a lot of unhappy experiences that might otherwise reduce your love for your fellow man.

Elementary Worldly Wisdom

The great lesson in microeconomics is to discriminate between when technology is going to help you and when it’s going to kill you. And most people do not get this straight in their heads. But a fellow like Buffett does.

For example, when we were in the textile business, which is a terrible commodity business, we were making low-end textiles—which are a real commodity product. And one day, the people came to Warren and said, “They’ve invented a new loom that we think will do twice as much work as our old ones

Advice to new Doctors starting practice

Hi all,

Please read the Disclaimers at the end of the post first, if you’re easily offended.


Generalists(general medicine):

  1. Get extremely unbeatable at 20 Questions(rationality link). It’ll help you make your initial diagnoses(ones based on questions about symptoms) faster and more accurate.
  2. Understand probability, bayes theorem and how to apply it** This will help you interpret the test results, you ordered based on the 20 questions.
  3.  Understand base rate fallacy, and how to avoid  being over confident.
  4. Understand the upsides and downsides of the drugs you prescribe. Know the probabilities of fatal and adverse side-effects and update them with evidence(Bayes’ theorem mentioned above) as you try out different brands and combinations.
  5. Know the costs and benefits of any treatment and help the patient make a good decision based on the cost-benefit analysis of treatment combined with the probabilities of outcome.
  6. Ask and Keep a history of medical records and allergies of the patient and till their grand parents.<sup>*</sup>
  7. Be willing and able to judge, when a patient is better off with a specialist. Try to keep in touch with Doctors nearby and hopeful all types of specialists.
  8. Explain the treatment options and pros and cons in easy language to the patients. It’ll reduce misunderstandings and eventually dis-satisfaction with the treatment.
  9. Resist the urge to treat patients as NPCs. Involve them in the treatment process.
  10. Meditate
  11. Find a hobby, that you can keep improving on till the end of life.
  12. Be aware of the conflict of interest between the patient and the pharmaceutical companies.
  13. Have enough research skills to form opinions on base rates/probabilities in different diseases and treatment methods as needed.


Basically the same skill sets as above. One difference is in the skill level and you should customize that as needed.

  1. For ex: You would need to be able to explain the treatment options and the probabilistic nature of the outcomes to your patients.
  2. As for research, keep a track of progress in your area in treatment methods and different outcomes on the “quality of life” for the patients after the treatment.
  3. Better applied Bayesian skills. In the sense of figuring out independent variables and their probabilities affecting the outcome.


Some controversial ideas(Better use your common-sense before trying out):

  1.  Experiment a little with your bio-chemistry and see how they affect your thought-processes.  To be safe, stick to biologically produced ones. For ex: injecting self with a small adrenalin dose and monitoring bodily response can help keep your thinking clear in emergency situations.
  2. Know your self biology better. For ex: male vs female differences mean the adrenalin response is different and peaks later in females.  If you think that’s wrong, please go back and check your course work. Also watch this 2 hour video and come back with objections after reading the studies he quotes.
  3. Keep regularly(whatever frequency your practice and nature of work demands) checking your(for ex; hormone levels) blood states, so that you  can start regulating your self for optimal decision-making skills.
  4. If you’re a woman, you’ll customize practice on some of the skill sets above differently. For ex: Mastery over emotions might need more practice, while empathizing/connecting with the patient might be easier.


  1. Most of what follows is based on my experiences(either as a patient myself or a concerned relative) with Indian Doctors. Some of it may be trivial, to others, but most of it is skills a doc will need and ignored in school.
  2.  I’ve split it in two (specialists and generalists) but there’s a fair amount of overlap.
  3. These are fairly high standards, but worth shooting for and I’ve kept the focus on smart rather than hard work.
  4. I’ve stayed from a few topics like: bedside manners/social skills, specific medical treatments and conditions(obviously, I’m not a Doctor after all) and a few others, you can add/delete(also specify/pick levels) as you see fit.
  5. Pick the skill-levels as demanded by your client population and adjust.
  6. I’m assuming generalists, don’t have to deal with emergency cases, but in some parts, that’s not likely then pick common emergency areas and follow specialist advice.
  7. I wrote this based on my experiences and with humans in mind, but veterinary Doctors may find some useful too.

* — I understand this is difficult in Indian circumstances, but I’ve seen it being done manually(simply leaves of prescriptions organized alphabetically, link to dr.rathinavel) , so it’s possible and worth the effort unless, you practice in area of highly migratory population.(for example rural vs urban areas).

**– If you’re trying to compete on availability for consultation, you’ll need to be able to do this after being woken in the middle of the night.

Notes/Thoughts from waiting at the labour ward.

  1. Child-birth or labour involves physical trauma, but it is expected to happen (for say 9 months)… that complicates things???? (It definitely breeds paranoia and worry-downward-spiral)
  2. However, the fact that it is expected and the fact that it is linked to species survival has facilitated a lot of research. I tend to think expected and predictable events get a bit over-researched and over-operationalized* in general. Is the fact that it is linked to reproduction causes too much(harmful) research or not?? Interesting question, but may not be answerable…
  3. The choice of word “labour” is interesting. It has marxist/communist connotations in most contexts, but in this, not really…. coincidence???
  4. The blanket ban of “no males in labour ward”** might be useful and valid, but it hardly is without exception and definitely allows for more follow-the-process decisions, encouraging a ignorance of context..
  5. Even after about 30 hours of partial sleep and adrenaline surge*** I could feel that jump in heart beat on looking at my daughter..
  6. P.S: every note above(except 5th) was written before the adrenaline surge.

* — to the point of over-engineering that risks violating the “do-no-evil”

**–in India

***–which was another story, i’ll write after my lymbic system settles down(aka down-regulating-lymbic-system for that memory), suffice it to say, i became more convinced of gender-difference (aka sexist)
Context location: