[Deprecated] Real Analysis Lecture 6.2 Continuity: Epsilon-Delta Definition

00:00 Recap
02:10 The Epsilon-Delta Definition
09:38 Continuity is a Local Property

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#RealAnalysis

[the_hidden_library]

1) Errata: At 09:00 it should be "|x-x_0| < delta" rather than "|x-x_0| < epsilon." (Thanks to @Aakash Yadav for the correction).

2) In the last section (starting at 09:38) I did not define the notion of a neighborhood (I do that in the next lecture). But here is the definition:

Let x be a real. A neighborhood of x is any open set that contains x.

In other words, a neighborhood of x is a subset U of the reals which contains x and has the property that if p is any point in U, then there is an open interval T containing p such that T is contained in U.

In yet other words, U is a neighborhood of x if U contains x and U can be expressed as a union of open intervals.

Thank you for pointing this out. (Thanks to @Aakash Yadav for pointing out this expository gap).

[aakashyadav]

At 09:00, it is |x - x_0| < delta and not epsilon.

[mathsfeeder]

I am a bit confused with calculus events like in high school we are taught everything with the help of limits...And if we go historically, even before limits mathematicians used "infinitesimals" which was obviously flawed so we used "limits" but then in real analysis, we see that we dont even use limits the way we use there.

Like high school definition of limit is dynamic that if x tends to a f(x) tends to L..But here we are like using a static definition of Limits that u chose a particular interval of x values and we wud give u that interval for f(x)..Basically now we are talking abt static points/ intervals unlike earlier where we were saying that out x is "approaching" a, we used to be taught like a arrow which is moving closer and closer to a but not reaching a.

Why in high school we werent taught with the static definiton, what was the problem with the simple dynamic definition bcuz that definition also quite beautifully omits the use of "infinitesimals"? like who decides that high school students shudnt be taught like that also is there any further change in the definition of limit or is there any other advancement as well?

[aakashyadav]

Doubt:
In the "Continuity is a Local Property" section, what do we mean by the neighborhood of x_0? Do we mean restricting the domain to (x_0-delta, x_0 + delta) for some positive delta?