Is There a Non-Constant Periodic Function Under Dilation?

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SUMMARY

The discussion centers on the existence of a non-constant periodic function under dilation, specifically questioning if a function f can satisfy the condition f(kx) = f(x) for a positive real number k. Participants agree that this concept, termed 'dilation periodicity', parallels traditional periodic functions defined by f(x+k) = f(x). The conversation encourages exploring substitutions to understand this property further.

PREREQUISITES
  • Understanding of periodic functions and their properties
  • Familiarity with the concept of dilation in mathematical functions
  • Basic knowledge of function transformations
  • Experience with mathematical substitutions and their implications
NEXT STEPS
  • Research the properties of dilation periodicity in mathematical functions
  • Explore examples of periodic functions and their transformations
  • Investigate the implications of substitutions in function analysis
  • Study advanced topics in functional equations and their applications
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Mathematicians, students of advanced calculus, and anyone interested in the properties of periodic functions and function transformations.

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is it possible to find a function f different from [tex]f(x)=constant[/tex]

with the property [tex]f(kx)=f(x)[/tex] for some real and positive 'k' ?

this is somehow 'dilation periodicity' is the equivalent to the periodic funciton [tex]f(x+k)=f(x)[/tex] for some positive 'k' for the traslation group
 
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yes

try a substitution :wink:​
 

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