#SoME2
This video gives a detailed construction of transition function for various levels of smoothness.

Sketch of proofs for 4 theorems regarding smoothness:
https://kaba.hilvi.org/homepage/blog/...

Faà di Bruno's formula:
https://en.wikipedia.org/wiki/Fa%C3%A...

Proof that e^(1/x) is smooth:
https://en.wikipedia.org/wiki/Nonana...

Chapters:
00:00 Intro
00:10 Definition of Smoothness
01:29 Hermite Interpolating Polynomials
04:32 Step Function
05:56 Auxiliary Theorems
08:43 Constructing Smooth Step Function
13:30 Main Problem
13:51 Outro

Music provided by BGM President
Track : Some Vintage Mood 7    • [Royalty Free Music] Some Vintage Moo...  

Corrections:
08:04 for the first theorem, f should be f'
08:15 for (2), it should be f(x) ≠ 0