Is there an algorithm to read this mathematical definition?

Question

I'm reading Gemignami's Calculus and Statistics.

I'm a little stuck in this definition:

Definition 7: Suppose $f$ is a function from a set $T$ of real numbers into $R$. Then $f$ is said to be continuous at a point $a$ of $T$ if, given any positive number $p$, there is a positive number $q$ such that whenever $x$ is a point of $T$ and $|x-a|<q$, then $|f(x)-f(a)|<p$. The function $f$ is said to be continuous if it is continuous at every point of $T$.

The question may seem naive, but I really miss an algorithm to read this definition. I usually need to look at examples, analogies, wikipedia and other sources to form a intuitive conception of the concept, and I guess that definition should be self-contained (right?). The problem for me is that there are a lot of premises that are assumed at the same time and I feel confused with that. I am unable to figure out what's happening in such definitions and to make things a little worse, I'm learning alone at the moment. So, Is there an algorithm/way to read this mathematical definition?

Answer 1

To me, the big thing is to figure out how to translate the definition in to English sentences that are not as precise, but are more telling.

The big, unsatisfying thing about my answer is this: in order to do this, you need to have some sense of what the definition is trying to tell you. But, I think, it doesn't have to be a very LARGE sense; plus, this is something that you get much better at as time passes. A lot of it depends on having access to a teacher who really understands it themselves (or, a community like this one, where there are lots of people who understand!)

For instance:

Given any definition of 'small' (given any positive number $p$)

we can define 'close' (there is a positive number $q$)

such that whenever $x$ is close to $a$, (such that whenever $x$ is a point of $T$ and $\lvert x-a\rvert<q$,)

the difference between $f(x)$ and $f(a)$ is small. ($\lvert f(x)-f(a)\rvert<p$.)

Answer 2

The idea here (of continuity at a point) is that we can keep $f(x)$ as close as we like to $f(a)$ (within a distance of $p$, however small) simply by keeping $x$ close enough to $a$ (within some distance $q$, which will depend on how large $p$ is, usually). You may also see this with $\epsilon$ instead of $p$ and $\delta$ instead of $q$, but it is the same idea, regardless.

For continuity of $f:T\to\Bbb R$, we can do this for every point $a$ in $T$, though in general, $q$ will depend on how large $p$ is and which $a$ we're looking at.

Answer 3

I like Cameron's answer, but I have one thing to add:

If you know about limits already, there's a nice way of phrasing this definition: The function is continuous at point $(a, f(a))$ iff: $$\lim_{x\to a}f(x) = f(a)$$

The function is continuous if it satisfies the above for all $x$ in the domain.

Of course, there's some qualifiers, etc. (e.g. "Let $f$ be a function from blaa blaa blaa...") for this definition, but you can think of continuity this way and be pretty safe.