Chapter 4 Continuity vs. discontinuity

This chapter is about functions. Well, convergence is about functions as well, functions on the positive numbers:

We used the notation $x_n$, but it actually means that the sequence (that is a function on the positive integers) takes the value $x_n$ on the integer $n$. Can you write $x(n)$ or $f(n)$ instead of $x_n$? Of course you can, and in some cases it looks quite neat. In this chapter however, we will study functions on real line, that is things in the form of $f:ℝ\to ℝ.$ Yes, like $\sin$, $\exp$ or the polynomials. Technically, the sin function should be written like that

but if you write just $\sin$ or even $\sin (x)$ it is still OK. You should remember that the notation $\sin (x)$ for a function is a bit controversial. It could mean the value of the function $\sin$ at the real number $x$, nevertheless we consider it a legitimate function notation. In case of the polynomials, $x^n$ seems to be a quite cute notation. (the absolutely correct notation would be: $P$, where $P(x)=x^n$ if $x\in ℝ$) If we want to restrict the domain of a function, that is, to consider $\sin :[0,1]\to ℝ$ instead of $\sin :ℝ\to ℝ$ (technically, these two functions differ!!!) we will make a comment.

Just to make sure…. the “picture” of the function sin, is not the function, it is the GRAPH of the function $\sin$, which is a subset of the plane that looks like a wave.