Write any two functions $$f \colon \mathbb{R} \rightarrow \mathbb{R}$$ such that $$f(x) = f\left(\frac{x}{2}\right)$$ for every real number $$x$$.

Correct Answer: 0

Solution

PLACEHOLDER ANSWER - the answer is two functions. Any constant function works, for example $$f(x) = 0$$ and $$f(x) = 5$$ for all $$x$$. A non-constant example is $$f(x) = 1$$ for $$x > 0$$, $$f(0) = 0$$ and $$f(x) = -1$$ for $$x < 0$$, since halving does not change the sign of $$x$$.

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