Elementary question on composition of functions

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SUMMARY

The composition of functions, specifically ##g \circ f## where ##f: R^n \rightarrow R^m## and ##g: R^m \rightarrow R^e##, is of class ##C^m## if both functions are class ##C^m##. This conclusion is established using the chain rule, which is a fundamental concept in calculus. The exercise is referenced in "Analysis on Manifolds" by James Munkres, providing a structured approach to understanding this topic.

PREREQUISITES
  • Understanding of class ##C^m## functions
  • Familiarity with the chain rule in calculus
  • Basic knowledge of function composition
  • Experience with real analysis concepts
NEXT STEPS
  • Study the chain rule in depth, focusing on its applications in real analysis
  • Read "Analysis on Manifolds" by James Munkres for comprehensive examples
  • Explore the properties of class ##C^m## functions and their implications
  • Practice problems involving function composition and differentiability
USEFUL FOR

Students of mathematics, particularly those studying real analysis, educators teaching calculus concepts, and anyone interested in the properties of differentiable 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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