Epsilontic – Limits and Continuity

[fresh_42]

Abstract

I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are calculated. At university, I soon met a quantity called epsilon. Algorithms became almost obsolete. They used $\varepsilon$ constantly but all we got to know was that it is a positive real number. Some said it was small but nobody said how small or small compared to what. This article is meant to introduce the reader to a world named epsilontic. There is quite a bit to say and I don’t want to bore readers with theoretical explanations. I will therefore try to explain the two basic subjects, continuity and limits, in the first two sections in terms a high school student can understand, and continue with the theoretical considerations afterward.
Limits
One of the first concepts in a calculus class is sequences. They are...

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[nuuskur]

I remember it took me a while to realise if a nonnegative number is smaller than any positive number, then it would have to be zero. Then I started understanding sandwiching arguments of the form
<br /> 0\leqslant |f(x)-f(a)| \leqslant g(x) \xrightarrow[x\to a]{}0.<br />
It's so obvious in retrospect, how on earth could I not understand something this simple..?

 

[fresh_42]

I remember it took me a while to realise if a nonnegative number is smaller than any positive number, then it would have to be zero. Then I started understanding sandwiching arguments of the form
<br /> 0\leqslant |f(x)-f(a)| \leqslant g(x) \xrightarrow[x\to a]{}0.<br />
It's so obvious in retrospect, how on earth could I not understand something this simple..?

I think there is a general difficulty that students have to overcome when they take the step from school to university. Perspective changes from algorithmic solutions toward proofs, techniques are new, and all of it is explained at a much higher speed, if at all since you can read it by yourself in a book, no repetitions, no algorithms. There are so many new impressions and rituals that it is hard to keep up. I still draw this picture of a discontinuous function in the article sometimes to sort out the qualifiers in the definition when I want to make sure to make no mistake: which comes first and which is variable, which depends on which.