Real Analysis | Showing a function is not uniformly continuous.

We prove a Theorem which allows us to easily show that a function is not uniformly continuous.

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[eberavilamartinez]

I'm revisiting analysis and this is a great channel to follow in the process.

[louisreinitz]

should have been |3n-3/n+1/n^3| at the end there.

[aliberro]

One of the best youtube channels, digging the new haircut ❤️

[matthartley]

You should have used xn = n+1/n, yn = n. You have a sign error at the very end, and your conclusion was that 1^3-0^3 > 3

[laurensiusfabianussteven]

yess, finally
finishing off x^3 for real, so satisfying

[사기꾼진우야내가죽여]

What we can notice from this proof is that when we negate the definition we can construct sequence since delta derived from the negation is an arbitrary posive number.

[gustavocardenas]

How is it that youtube did not suggest me your channel before!(? ) Great video, thanks a lot!

[senhueichen]

Nice haircut, as nice as Dedekind cut.

[kevinmartin]

I was going to ask if uniform continuity means the same as "continuous everywhere" but clearly it does not.


That final proof (for f(x)=x^3 not being uniformly continuous on A does not work if A has finite limits, since you can no longer generate the two infinite sequences {x_n} and {y_n}.

Essentially, the problem stems from the "Application: f(x)=x^3 is not uniformly continuous" which should have specified that the domain A is the entire set of real numbers. The property of being uniformly continuous requires specification of the domain to be meaningful. I suppose one could argue that by not specifying the domain, it is implicitly the real numbers, but given that up to that point the discussion involved a specific domain A, suddenly changing to an implied domain requires at least a comment to that effect. Or perhaps this is some carry-over context from the previous video where he was unable to show uniform continuity for f(x)=x^3.


It looks sort of like uniform continuity over A requires that the first derivative be finite in A. Otherwise one can always generate a sequence pair as was done here, except instead of approaching infinity, the sequences would approach a point where the first derivative becomes infinite.

[CM_France]

Hi,

Could you please rise up a little bit your camera? When a sub-title appears on the top of the screen, it may hide what you wrote. And the concrete band in the bottom is not usefull. Thanks.

[AMOODHG]

Hello all! I don't get the part why we've set delta to be 1/n?

[utsav]

Please help me in this question
If each of the algebraic expressions
lx^2 + mx + n
mx^2 + nx + l
nx^2 + lx + m
Are perfect squares, then proove that (l+m)/n=-4
Plz help😢😢

[TheMauror]

So, is there a connection between the derivative and uniform continuity?