Question

Consider those continuous functions \( f : \mathbb{R} \to \mathbb{R} \) that have the property that for every \(x \in \mathbb{R},\) \[ f(x) \in \mathbb{Q} \text{ if and only if } f(x + 1) \notin \mathbb{Q}. \] The number of such functions is _________.

Hint:

A continuous function on \(\mathbb{R}\) cannot alternate between rational and irrational values because the preimage of \(\mathbb{Q}\) under a continuous map cannot be dense and disjoint.

Answer 1

Correct Answer: 0

Step 1: Understanding the condition.
We are told \(f(x) \in \mathbb{Q}\) iff \(f(x + 1) \notin \mathbb{Q}.\) So rational and irrational values alternate for \(x\) and \(x+1.\)
Step 2: Check continuity.
The set of rational and irrational numbers are both dense in \(\mathbb{R}.\) Hence, such alternating behavior is impossible for a continuous function — it would require discontinuous jumps between rational and irrational values.
Step 3: Conclusion.
No continuous function satisfies the given property. Hence, the number of such functions is \(\boxed{0}.\)