Prove the Piecewise Function is Continuous with the Delta-Epsilon Definition of Continuity

In this video we prove that this piecewise function is continuous at x = 0. To do this we use the delta-epsilon definition of continuity.

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[kingbeauregard]

Epsilon-delta tends to make people run screaming, and after turning it over and over in my head for a long time, I think the issue is, people can't visualize what is going on, so they don't quite have a sense of where they're going with the math.

Let's say you're trying to prove the limit of a function f is L at x=c. So, imagine you're drawing a rectangle centered at the point (c, L), and it is of dimensions such that the function f never bumps into the top or bottom edges of the rectangle. Can you keep drawing smaller and smaller rectangles, down to no height and no width, such that the function f never hits the top or bottom edges? If you can do that, then you've proven the limit.

So math-wise, what we want to do is come up with a mathematical relationship between the height and width of our rectangles: we want to establish that, for every height, there will be a width that gives us a rectangle where the function doesn't hit the top or bottom edges. Our height is 2*epsilon (from L - epsilon to L + epsilon), our width is 2*delta (from c - delta to c + delta).

The process is to start with our initial function f, and come up with another function that is easier to deal with than f, that we can also mathematically prove is further away from L than our original function f was (so it's "bigger", if you will). And if that function isn't easy enough to deal with, then we come up with another function that's both "bigger" and easier to deal with. Every time we do this, we are requiring that epsilon get bigger and bigger, which is fine: at the end of this we hope to arrive at some simple relationship between epsilon and delta that scales all the way down to nothing. Since we will have established that our original function f is "smaller" than every intermediate function and they're all "smaller" than our simple epsilon-delta relationship ... well, since that epsilon-delta relationship scales all the way down to nothing, so must the original function f.

A lot of the time, you make the process simpler by establishing an arbitrary and convenient cap on how wide your rectangles will be. That allows you to say, with great certainty, that your original function f is "smaller" than a straight line in that region, which means you are just about done (you just have to write delta as a function of epsilon in that straight-line region). In math terms, you will define delta as the minimum of that function of epsilon, and the arbitrary and convenient cap.

[neelshah]

How can we prove that this function is differentiable also using epsilon-delta ??

[algorithme]

x belongs to R what if x = 0 ? f(x) = 0 then ?

[despicable]

at 3:16 can we write |sinx| <= x (instead of |sinx|<=1)
and after switching sin(1/x) with 1/x,
the formula will be |x^2 * 1/x| <Epsilon
thus |x|<Epsilon which is equal to delta?
@TheMathSorcerer