Friday, 19 April 2013

real analysis - Function Satisfying f(x)=f(2x+1)



If f:RR is a continuous function and satisfies f(x)=f(2x+1), then its not to hard to show that f is a constant.




My question is suppose f is continuous and it satisfies f(x)=f(2x+1), then can the domain of f be restricted so that f doesn't remain a constant. If yes, then give an example of such a function.


Answer



Let f have value 1 on [0,) and value 0 on (,1]. This function is not constant (although it is locally constant), and satisfies f(x)=f(2x+1) whenever x is in its domain.


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