#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/differentiable.htm Faà di Bruno's formula: https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula Proof that e^(-1/x) is smooth: https://en.wikipedia.org/wiki/Non-analytic_smooth_function 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 - https://www.youtube.com/watch?v=GU8I6g1Fph4 Corrections: 08:04 for the first theorem, f should be f' 08:15 for (2), it should be f(x) ≠ 0