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.

Squeeze theorem

To quote from “The girl next door”
The first lesson of politics is “Always know whether the squeeze is worth the juice”. Now i was trying to finally make a genuine effort at understanding Central Limit theorem. Throughout my life(30 years), i have always been suspicious whenever statistics goes beyond the mean, median, mode, SD and Variance. (i.e to say, whenever any stat goes above first and second moments). Part of it because i never really learnt or rather never paid enough attention to convince myself of the theorems involved in reasoning with distributions. Anyways, i figured Central limit theorem would be a good place to start and in learning by teaching am summarizing what i’ve learnt so far.

It started off as i came across this post on HN and going through comments and critique realized the demo is more of a special case and while i did get that specific example(and sure of what CLT says) am still unsure of why Central limit theorem is true or how one formulate it in math terms. It is important for me to understand those, if i am ever to be able to question someone claiming some implication of CLT. Anyway, i came across the squeeze theorem in one of the HN comments and since it seems it’s part of the proof for CLT, I ended up reading and here’s the result of that.

Anyway, enough story. Let’s go onwards. So here goes straight from the wikipedia page:

Assumptions:
There are three functions f,g,h defined over a limit l.
a is a limit point.
f,g,h may not be defined at a, since it is the limit point.

g(x) leq f(x) leq h(x)

lim_{x to a} g(x) = lim_{x to a} h(x) = L

To be proved:
lim_{x to a} f(x) = L

Proof:

Limits:

I’ll try and clarify what is a limit as mathematically defined, and hopefully without equations,but words only.
Well, according to wikipedia page, limit of a function f(x) means that the function f(x) can be made as close to a value (say L),
by making x sufficiently close to c.

Or to write out the equation
lim_{x to c}f(x) = L