I was reading this definition and I have always been stuck as to what it means in words. Could someone explain what the function below means in words?
$f:[a,b]\rightarrow \mathbb{R}$ is a continuous function $(*)$
So what exactly is $[a,b]$? How would I interpret/explain it in words?
i.e. I know that $f:X\rightarrow Y$ can be interpreted as saying "Let $f$ be a function from $X$ to $Y$". I'm just not sure how to do it for $*$.
Thanks.
Best Answer
$[a,b]$ means the closed interval from $a$ to $b$, where $a$ and $b$ are elements of some ordered set, usually $\mathbb R$, in which case it's $\{x\in\mathbb R|a\le x\le b\}$. (But the notation is used also in $\mathbb R$-order trees, for example, and other sets.)
$f:[a,b]\to\mathbb R$ means $f$ is a function from $[a,b]$ to $\mathbb R$; it's continuous if its value at any point $c\in[a,b]$ is the limit of its values at $x\in[a,b]$ as $x\to c$.