In this video, I will present a VERY important and useful theorem
about continuity.
It is so useful that you have probably already used it,
even if you had not realized it.
As a reminder, here is the definition of continuity:
a function f is continuous at a point
when calculating the limit is the same as evaluating.
Not all functions are continuous everywhere.
If they were, computing limits would be much easier!
But a LOT of common functions ARE continuous,
and THAT is what this theorem says.
Specifically, any function we can construct
with sum, product, quotient, and composition
of polynomials, roots, trigonometric functions, exponentials,
logarithms, and absolute values is continuous -- on its domain.
This is a lot of functions,
and they are the functions that we use most often.
Sometimes, these functions are called “elementary functions”.
“Elementary” does not mean “easy”; it is just the name for this class.
For example, how do we compute
the limit, as x approaches 1, of this ugly-looking function?
As complex as the function may look,
it is in the class of functions in the theorem, so it is continuous.
1 is certainly on its domain.
So to compute the limit, I simply plug in x = 1.
That’s it. Nice!
An important warning:
the theorem ONLY applies to these functions ON THEIR DOMAIN.
So NOT all limits involving these functions
are a matter of “plug in the value.”
For example,
the function f(x) = sin(x)/x is NOT continuous at x = 0,
because 0 is NOT in the domain.
If I wanted to compute its limit as x approaches 0,
I could NOT simply evaluate.
Or the function tangent is not continuous at x = π/2,
because π/2 is NOT in the domain.
But still, this theorem is extremely useful.
It is worth proving it.
Of course, I cannot prove it for all functions one by one.
Instead, I have a better plan. The plan goes in two steps.
Step 1: I need to prove that the “basic” functions are continuous.
The "basic" functions
are the building blocks that I use to construct other functions.
I have chosen 6 "basic" functions:
constant functions, the function f(x) = x,
basic exponential, natural logarithm, sine, and absolute value.
Step 2: I need to prove that the sum, product, quotient,
and composition of continuous functions are continuous.
And that will be it.
Perhaps you think that I forgot some basic functions,
but I didn’t; these are enough.
Other functions that we might have considered basic
can be BUILT from these ones.
For example,
cosine can be written as the composition of sine and a polynomial.
Or tangent can be written as the quotient of sine and cosine.
What about exponentials with other bases? Or square roots?
They can both be written
as the composition of e^x and other functions.
Okay, then. We need to take these two steps.
I am not going to prove
that the 6 basic functions are continuous.
I hope I have persuaded you that I have simplified the problem enough
by reducing it to only 6.
They can be done; some are easier than others,
but I will not write those proofs for now.
What about step 2?
Use the limit laws!
For example, I will prove
that the sum of continuous functions is continuous.
Let f and g be two continuous functions at a.
By definition, that means that their limits as x approaches a
are simply f(a) and g(a).
Then the limit law for sums
tells me that the limit of f + g will be the sum of the limits:
f(a) + g(a).
But by definition, this means that f + g is continuous at a.
And that completes the proof; that’s it!
That was easy.
And the exact same argument will work
for the product and the quotient of continuous functions,
using the corresponding limit laws.
Then, what is left to prove?
We still need to prove
that the composition of continuous functions is continuous.
The problem
is that we do NOT have a limit law for composition.
THAT is an interesting problem on its own,
and it is the topic of the NEXT video.
In it, I will explain
why there ISN'T a limit law for composition,
but nevertheless,
composition of continuous functions IS still continuous.