Real Analysis 27 | Continuity and Examples

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:18 Definition Continuity
01:22 Special case (holes in the domain of definition)
02:56 Extended definition of continuity
03:48 Examples
09:28 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[brockobama]

The Bright Side of My Grade that will plummet after my midterm today.

[douglasstrother]

My Calculus Professor (Tony Tromba, UC Santa Cruz, Fall 1981) dropped the last example on us at the end of a Friday lecture to give something to snack on during Happy Hour.

[angelmendez-rivera]

One characterization I personally prefer, in the case where x0 is not an isolated point of the domain, is where continuity of f at x0 is defined true if and only if lim f(x) – f(x0) (x —> x0) = 0. This superficially may seem like an unnecessarily complicated way of characterizing continuity, but this is actually a very useful and extremely elegant characterization, because it makes for an intuitive and simple segue into uniform continuity, making the connection between continuity everywhere and uniform continuity almost trivial, and analogous to the connection between pointwise convergence everywhere and uniform convergence. Unexpectedly, it also creates a very nice segue into defining differentiability later on, and other types of continuity, such as Lipschitz continuity.

[filmmyduniya-mfhq]

When we talking about differentiability it is easy to under this is defined on an open interval because in the definition of derivative f(x+h) defined for only interior point but why we use closed interval in tge continuity definition. Same problem occour here also?

[okikiolaotitoloju]

Just to confirm at 2:33. you are saying that if x0 is isolated, such that there is nothing around there then the function is not continuous at x0.

[Ok-egdg]

For |x| why does limit to infinity equal 0 and not infinity?

[anotheperspective]

Why is the lim x = 0 when n —>inf? I thought it will be infinity.?😰

[ffar]

4:28 you first say that the left limit is different from the right limit. Then, you say that the limit on the point does simply not exist. So, which limits exist now and what is their value? I see it rather that the limits exist but the one is different from the function value at 0.

[MrWater]

Could be possible that in the page 57 is there a mistake where "Then f (xn) = 0 for all n ∈ N and thus limn→∞ xn = 0 != f(x0) = 1" why is the limn→∞ xn = 0 instead of limn→∞ f(xn) = 0? (same question for the second case? Thanks you again!

[carl]

Using the notion of 'density' (of Q in R) in a real analysis introductory course without previously explaining it seems a bit abrupt.

[erlint]

Isn't x ∉ Q too broad? As in it includes all numbers that are not in Q including complex? Wouldn't it be more precise to use x ∈ R\Q?