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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 $*$.


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Should we combine this question with… – Zev Chonoles Jan 1 '12 at 23:07
up vote 5 down vote accepted

$[a,b]$ 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 $x\to c$.

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$[a,b]$ consists of all the numbers greater than or equal to $a$ which are also less than or equal to $b$.

If $b$ is less than $a$, $[a,b]$ is empty.

If $a$ and $b$ are equal, $[a,b]$ is a single point equal to $a$ (or $b$).

If $b$ is greater than $a$, $[a,b]$ is a line segment on the real line between $a$ and $b$ with the endpoints included.

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Given $f:[a,b]\rightarrow \mathbb{R}$, the interval $[a,b]$ is the domain of the function. It a closed interval which contains all the values for $x$ between and including $a$ and $b$.

So if the function is continuous on this interval what we mean is that for every $c\in[a,b]$, $f(c)$ exists and is equal to the limit as $x\rightarrow c$.

Thus, for every $c\in [a,b], \lim_{x\rightarrow c}f(x)=f(c)$.

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