Stretching = Translation

Algebra Level 2

\[\alpha f(x) = f(x + \alpha) \]

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


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 \( 0 < \alpha < 1\)? What happens when \( \alpha < 0 \)?

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