In this video,
I will illustrate with an example how to write a rigorous definition.
My goal is to define "increasing function."
I have chosen this concept,
because we all have some kind of geometric intuition behind it;
we know what it means, more or less:
it means a function that always goes up, like the one in this graph.
But saying "it always goes up" doesn't count as a definition.
So, it's not that this content is particularly important,
but it is going to help me illustrate the difference
between having an idea of what the concept means more or less
and what we'll count as a valid, precise, rigorous, mathematical, definition.
Some students like to think
that the definition of "increasing" is positive derivative.
That is wrong;
having a positive derivative is RELATED to being increasing,
but is not quite the same thing.
Here is an example:
the function in this the graph is definitely increasing; it always goes up.
But at this point, it has a corner. And at the corner, it has no derivative.
So if I use "positive derivative" as a definition of "increasing,"
then I have to conclude that this function is NOT increasing.
That is a bad definition.
I want my definition of increasing
to include ALL the functions that I want to be "increasing,"
even this one.
So let's forget this, and try instead
to come up with a definition that doesn't use derivatives at all
To formalize the idea of "the function always goes up,"
I'm going to compare what happens to the function at two values of x,
which I'm going to call x_1 and x_2.
And in this picture,
I notice that x_1 is smaller than x_2, and f(x_1) is smaller than f(x_2).
That's a little bit of progress, but I cannot say
that these two inequalities by themselves are the definition;
I need context.
I need to explain what is f, what is x_1 and what is x_2.
So let's write this a little bit better.
I'm going to say
that a function f is increasing when [something related to this].
And now, I have introduced f.
What about x_1 and x_2?
x_1 and x_2 should be values in the domain of f.
And actually,
the domain of f is very important
because I could have a function that is increasing on an interval,
and then decreasing on another one.
So let's give a name to that domain.
I'm going to say
that a function f is increasing on an interval I when,
and now I have these inequalities, where x_1 and x_2 are in I.
And notice something important:
the concept I'm defining is not "increasing,"
it's "increasing on."
Again, because a function could be increasing ON one interval,
and NOT increasing ON another one.
Now, what I have written here is a little bit better,
but it's not quite enough yet;
as I wrote, it's meaningless.
Because what am I saying about these inequalities?
Do I want it to be true for SOME values of x_1 and x_2?
For ALL values? Something else?
Think about it.
For the function to be increasing,
I want to be able to compare every pair of elements x_1 and x_2.
So I need to write here a quantifier "for all,"
otherwise I don't know what this means.
This is better, but we're not done yet.
Now, at the very least,
this is a precise mathematical statement that means SOMETHING.
But this is NOT what I want.
Let's read it carefully.
At the moment, it says that for EVERY x_1 and x_2,
x_1 is less than x_2, AND f(x_1) is less than f(x_2).
That's what it says.
But that's not what I want; that's never true, right?
It is not true that EVERY pair of elements x_1 and x_2
satisfies even the FIRST inequality.
Actually, what I WANT to say is that for every pair of elements,
WHENEVER they satisfy the first inequality,
they must ALSO satisfy the second inequality.
Or in other words,
IF the first inequality is true, THEN the second inequality is true.
And what I have here is now a conditional.
And now, this has a precise meaning,
and in fact, this is the correct definition.
What I want to take from this example
is that every little piece I put here in the definition
wasn't just ME being a pedantic mathematician,
it was NECESSARY.
From introducing my notation
to noticing that I need to define increasing ON an interval,
rather than plain increasing,
to using the quantifier and the conditional,
all those things were necessary; without THEM,
this could mean something different, or not mean anything at all.
But with all these pieces put together,
the definition means precisely what I want it to mean and nothing else.