8 - A little about Continuous Functions

A function $y=f\left(x\right)$  is called a continuous function if little changes in the values of $x$  will generate little changes in the values of $y$. Suppose $b$  is a $y$   value and the corresponding $x$  value is $a$ , in other words, suppose $f\left(a\right)=b$. Then if $d$  is sufficiently close to $b$  , then we can find a real number $c$  sufficiently close to $a$  such that $f\left(c\right)=d$. In this case, we say that the function $f\left(x\right)$  is continuous at $x=a$.

As you may have guessed from the previous post about rational and irrational numbers, if $f\left(x\right)$  is a continuous function and if $q$  is an irrational number and ${\left\{{r}_{k}\right\}}_{k=1}^{k=\infty }$  is an infinite sequnce of rational numbers such that $\underset{k\to \infty }{\mathrm{lim}}{r}_{k}=q$  , then the infinite sequnce of real numbers ${\left\{f\left({r}_{k}\right)\right\}}_{k=1}^{k=\infty }$  will get very close to the real number $f\left(q\right)$. In other words, $\underset{k\to \infty }{\mathrm{lim}}f\left({r}_{k}\right)=f\left(q\right)$.

We use this technique of approximation or limiting process when we deal with irrational numbers and functions of real numbers. The big restriction is that the function must be continuous. Thankfully, functions are mostly continuous. And mostly, even when a given function is not continuous everywhere, the numbers at which it will be discontinous will be a few finite set of numbers. There are, however, some odd ball functions that are studied only for their oddities, that are discontinuous almost everywhere. But they are like totally sick jokes in real life.