Real Analysis 27 | Continuity and Examples

[@brockobama257]

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

[@angelmendez-rivera351]

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.

[@douglasstrother6584]

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.

[@brightsideofmaths]

Nice! It's a standard example in analysis :)

[@dyllanusher1379]

An interesting example is the thomaes function. It's a function that is continuous on the irrationals and discontinuous on the rationals. Another interesting example is a modified Dirichlet func, x maps to 0 of x not rational, x maps to x if irrational. So it's continuous at x=0 and discontinuous elsewhere.

[@okikiolaotitoloju2208]

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.

[@brightsideofmaths]

If x_0 is isolated, the function will be always continuous there.

[@JojiThomas7431]

Good video.

[@offthepathworks9171]

Lovely, just lovely.

[@filmmyduniya-mf1hq]

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?

[@brightsideofmaths]

Boundary points of an interval are not really a problem, also not for the derivative.

[@Hold_it]

Nice! 👍

[@RSA_Shock]

Nice video

[@bangprob]

Thanks

[@anotheperspective]

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

[@Cattooo-k3i]

I believe it means xn apporaches that 1 point infinitely which is x=0 point, ,and its 0 ( I guess, I am new to real analysis )

[@Ok-eg8dg]

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

[@brightsideofmaths]

It's the limit to zero.

[@ffar2981]

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.

[@brightsideofmaths]

The limit does not exist since the approximation from the left is different from the approximation from the right.

[@ffar2981]

@@brightsideofmaths Thanks for your quick reply! Do you mean limit from the left (right) when you say approximation from the left (right)? What is then the 'overall limit'?

[@brightsideofmaths]

@@ffar2981 Yes, limits from right and left. The overall limit is the actual limit.

[@ffar2981]

Okay, got it, thanks :-)

[@MrWater2]

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!

[@brightsideofmaths]

Page 57 means in my book?

[@MrWater2]

@@brightsideofmaths Yes!😉

[@brightsideofmaths]

Oh, you are right. I correct that :) Thanks!

[@carl3260]

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

[@brightsideofmaths]

Do you think so? I have my Start Learning Reals series where we exactly introduce this concept while defining the real numbers.

[@brightsideofmaths]

@Mr Fl0v I guess if you formulate both carefully in the correct way, it should be equivalent.

[@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?

[@brightsideofmaths]

Complex numbers with non-zero imaginary part are definitely not in Q.