Undergrad Elementary question on composition of functions

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The composition of functions g∘f, where f: R^n → R^m and g: R^m → R^e are both class C^m, is indeed of class C^m. This can be proven using the chain rule, which establishes that the differentiability of the composition follows from the differentiability of the individual functions. The discussion references Munkres' "Analysis on Manifolds" as a potential source for similar exercises. Understanding the application of the chain rule is crucial for confirming the composition's class. Overall, the composition retains the smoothness properties of the original functions.
facenian
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Helo, given ##f:R^n\rightarrow R^m## and ##g:R^m\rightarrow R^e## both class ##C^m##. Is the composition ##g\circ f## of class ##C^m## ?.
 
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Yes, use the chain rule to prove it.
 
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BTW, I believe this exercise appears in Munkres Analysis on Manifolds with a suitable guidance, but I am not 100% sure.
 

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