## Continuity – (Part Two).

Definition:

(i) A function $f$ is said to be continuous from the right at a if $\lim\limits_{x \to a^{+}} f(x) = f(a)$ We can see that, as the function approaches a certain x-value from the right, $f$ is defined and $\lim\limits_{x \to a^{+}} f(x) \equiv f(a)$

And as the function approaches a certain x-value from the left, $f$ is not defined, i.e; $\lim\limits_{x \to a^{-}} f(x) \neq f(a)$.

Therefore, we say that the function is continuous from the right at this point, but is discontinuous from the left.

(ii) A function $f$ is said to be continuous from the left at a if $\lim\limits_{x \to a^{-}} f(x) = f(a)$ Here, it is clear from the graph that the function is continuous from the left as approaches 3. This is because the function is defined at x = 3 and, $\lim\limits_{x \to 3^{-}} f(x) \equiv f(3)$

However, from the right, $\lim\limits_{x \to 3^{+}} f(x) \neq f(3)$

So, we can say that the function is continuous from the left, but discontinuous from the right, at the point x = 3.

We may also have a function where it is neither continuous at a point from the left or from the right but is defined elsewhere.…

## The 2016 International Earth and Sky Photo Contest

If you’re into astrophotography, take a look at this contest. How clear is this post?

## Continuity – (Part One).

Definition:

A function $f(x)$ is continuous at a given point x = a if those three conditions below are met “simultaneously”:

(i) $f(a)$ is defined. (i.e; a is in the domain of $f$)

(ii) $\lim\limits_{x \to a} f(x)$ exists.

(iii) $\lim\limits_{x \to a} f(x) = f(a).$

NOTE:

• If any one of the three conditions is false, then $f$ is discontinuous at a, or it has a discontinuity at a.

Let’s now look at the different cases where $f(x)$ may not be continuous at x = a.

(i) $f(a)$ is defined but $\lim\limits_{x \to a} f(x)$ does not exist. At a = 0, the function is not continuous despite $f(a)$ is defined (Here, $f(a)$ is equal to -1). This is because the two one-sided limits are not equal and as a consequence, the limit does not exist. This is called a jump discontinuity.

(ii) $\lim\limits_{x \to a} f(x)$ exists, but $f(a)$ is not defined. Assume a is the x-value where there is a hole in the graph. We can see that the limit from the right of a and the limit from the left of a are equal.…

## Do You Find Mathematics Scary?

A few weeks ago I attended a lecture by Johnathan Lewin, regarding the use of technology when teaching and it was brilliant, and I’m not even talking about his use of technology. The passion that Johnathan speaks with and the passion he has for Mathematics is explosive and practically contagious.

He uses a number of different programmes and applications to assist him in the classroom. He even records his lectures (he captures the audio and a visual of the learning materials and then makes them available to his students). He is in favour of designing the materials in front of the learners in order for them to see how the Mathematics is created rather than to arrive with some neatly prepared sides and show them what Mathematics looks like. He wants them to engage in it at all levels and not just see the perfect final product, if you wish.…

## Orthogonality and Volatility

In quantitive finance, there are always many thousands of simultaneous bets available. You can be short oil, long Apple and short the Yen all at the same time. If you have a good model of the assets, then that model will tell you that some of these bets have positive expected value.

Being faced with thousands of positive expected value bets sounds like Christmas but today it’s Easter. Positive expect value bets are only necessarily good when they are independent of the other bets you’re holding. This is because there’s a lot of math on your side if you can get your bets to be independent. In particular, the shape of the cumulative distribution function of the Binomial distribution shows how hard it is to lose when you place lots of independent bets. And also, the Kelly Criterion will tell you how much to risk on your independent bets.

Statistical or machine learning models will tend to produce portfolios with some bias.…

## Computational Complexity; Article 2

Beauty is in the Mind of the Beholder
Often mathematicians speak of finding beauty in their subject, it is reasonable to ask what they mean. Of course, one will get as many answers as there are maths practitioners, but I shall hazard a generalized answer here. Mathematicians deal with abstract objects that exist in the mind, and share a suspicion that these abstractions are in fact real. So like real things they have properties, and enter into relationships. The feeling is that if abstractions behave as real things, then they must themselves, at some level, be real. If they are real, then they must also have relations with other concrete things, and indeed, the ultimate aspiration is to develop an abstraction that describes the world. This is the fundamental quality that separates the practice of maths from, say, intellectual games. Our minds evolved to find order in the universe and mathematics gives us as systematic and dependable (to some extent) way of doing so.…

## Calculating the date of Easter, and a computational challenge

The date on which Easter Sunday falls was decided, by the Council of Nicaea in 325 AD, to be the first full Sunday after the first full moon after the Vernal Equinox. This is very easy to work out with a lunar calendar or if you have data about the lunar cycles and the equinoxes, but what if you just were given the year. How would you work out what date it was on? Well, it turns out that there isn’t a very simple formula which can tell you this, but there is a rather complicated formula. This was presented to the journal Nature in 1876, though I haven’t been able to track down the author of this particular formula. So, this is how you calculate the date of Easter Sunday:

1. Take the year (in the Gregorian calendar) and divide it by 19. This will give you a quotient and a remainder.

## Proof by contradiction – part 2

So, in the last post we proved that $\sqrt{2}$ is irrational, by trying to see what the consequences would be if it were rational. We first said that if it were rational, then we should be able to write it in a simplest form $\frac{p}{q}$ where p and q had no common factors, and then showed that in fact this was impossible, so our original proposition was indeed true (as trying to prove otherwise gave us a contradiction).
Now we are going to look at another example, which looks very similar, but here our contradiction will be a little different to last time. The fact is that, unlike much of what you would have done at highschool, there isn’t such a roadmap to how to do things here – you have to figure out for yourself where the contradiction comes in. In these posts I will point them out for you, but in general, you need to build your intuition about where something looks a bit dodgy and a contradiction is raising its head.…

## Proof by contradiction – part 1

Proof by contradiction may at first seem completely weird! I give you something to prove, and you seem to ignore me and try and prove that what I want you to prove is wrong!

Actually, this isn’t nearly as strange as it first seems, and it can work in contexts other than mathematics. The idea stems from the fact that a statement is either true, or false (well, if you listen to Gödel, then you have to be a bit careful, but it’s reasonable enough for now). The process is the following:

1. You want to prove that a statement is true.
2. You say “what would happen if the statement were actually false?”
3. You explore the consequences of it being false.
4. If you find that it gives you a contradiction (something which you claimed to be true, but which you now see isn’t true), then you know that in fact the original statement can’t be false…so it must be true, and you’ve proved it.