Stretching = Translation

Algebra Level 2

αf(x)=f(x+α)\alpha f(x) = f(x + \alpha)

Given that α \alpha is a real number greater than 1, does a non-constant function f ⁣:RRf \colon \mathbb{R} \to \mathbb{R} exist that satisfies the above functional equation for all x x ?


Bonus

  • If you think the answer is yes, then find all functions that satisfy the given equation.

  • If you think the answer is no, then prove it.

  • What happens when 0<α<1 0 < \alpha < 1? What happens when α<0 \alpha < 0 ?

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