# A Q-linear transformation!

Algebra Level 5

Let $$\alpha \in \mathbb{R}$$ fixed, and $$f:(\mathbb{R}, | \cdot |) \to (\mathbb{R}, | \cdot |)$$ be a function such that $f(\alpha(x + y)) = f(x) + f(y), \space \forall \space x, y \in \mathbb{R}$ Assuming the axiom of choice. How many of the following statements are always true?

a) $$f(0) = 0$$

b) $$f$$ is an odd function, i. e. $$f(x) = - f(-x), \space \forall \space x \in \mathbb{R}$$

c) If $$\alpha \neq 1$$ then $$f(x) = 0, \space \forall \space x \in \mathbb{R}$$ and therefore, $$f$$ is a continuous function.

d) If $$\alpha = 1$$ then $$f(x) = Cx$$ where $$C \in \mathbb{R}$$ is a constant.

e) If $$\alpha = 1$$ then $$f$$ is a continuous function.

Inspiration

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