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.



Problem Loading...

Note Loading...

Set Loading...