Do diffeomorphisms have to be one-to-one functions?

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SUMMARY

A diffeomorphism is defined as a function that has a differentiable inverse, necessitating that the original function be one-to-one. If a function is not one-to-one, it cannot possess an inverse that is also a function, as inverses must map each output back to a unique input. Therefore, the requirement for a function to be one-to-one is essential for it to qualify as a diffeomorphism.

PREREQUISITES
  • Understanding of differentiable functions
  • Knowledge of inverse functions
  • Familiarity with one-to-one functions
  • Basic concepts of calculus
NEXT STEPS
  • Study the properties of one-to-one functions in calculus
  • Learn about differentiable functions and their characteristics
  • Explore the concept of inverse functions in depth
  • Investigate the formal definition and examples of diffeomorphisms
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Students of calculus, mathematicians interested in topology, and anyone studying the properties of functions and their inverses.

Antineutrino
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The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?

Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
 
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Antineutrino said:
The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?

Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
If a function is not one-to-one, then it has no inverse.
 

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