# 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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