Real Analysis Ep 22: Thomae's function and epsilon-delta

[@sleepeasyguitar9505]

0:00 - Introduction 0:34 - D's function 1:35 - Thomae's function 19:54 - The limit of a function 26:20 - Epsilon-delta definition 34:30 - Examples

[@lainauser1961]

All your videos help me so much ! Many thanks !!

[@quant-prep2843]

my gifted professor!

[@DJ-nh3on]

We can't use unreduced form for rational number and apply the definition of Thomae's function. Moreover, how we can depend on approximation in proofing the continuity?!

[@ChrisStaecker]

Yes you're right- I said you need to write it in reduced form as part of the definition. This is not a problem though- every rational number has a unique reduced form, so there is no ambiguity or any issue in requiring the fraction to be reduced as part of the definition. My discussion of the continuity was not quite a rigorous proof, as you noticed. But the actual proof uses the same basic reasoning. For any real number x, we consider any sequence of rational number approximations q_1,q_2,... approaching x. Then these rationals must have increasing denominators (this detail is a hard part of the proof, but I tried to give the idea here), which means that t(q_n) -> 0. And this means that t is continuous at x. This class meeting was before we really talked about the definition of continuity, and before we did the sequential criterion for continuity, so I was being informal on the details. My goal was to show an example of a wild function and how the notion of continuity is more complicated than the "don't have to lift my pencil" explanation that kids get in lower-level classes.

[@hunterroy8485]

"Master of calculus in our presence here" lol

[@erictao8396]

I love your videos; they're so helpful! At 33:51, when you introduce the definition of limit of a function in terms of neighborhoods, should it be V_delta(c) - {c} instead of just V_delta(c), since we don't want to look at the value of f(x) at x=c? Also, is there an episode 23? Thank you for posting these videos on KZbin!!

[@ChrisStaecker]

Yes you are right about the neighborhoods! Thanks for paying close attention. We almost never used this neighborhood point of view in the course, so I wasn't careful enough when I said that part. Thanks! And there is an episode 23- somehow it was left off the playlist but I just put it in now. Thanks-

[@erictao8396]

@@ChrisStaecker Thanks so much for the quick reply!

[@louisb8672]

so good yeeeeee LOVE U

[@hunterroy8485]

Thomae's Function seems like a random intellectual construction built by a dude in the 1800s with a pointy beard and too much time on his hand......

[@ChrisStaecker]

Yes- not terribly "useful" in the ordinary sense, but to me it's a testament to how inadequate the "really really close" viewpoint of continuity is. Puts to shame all the folks who think the concept of continuity is "simple".