Function with period (!)

Give an example of a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that any rational number is its period but any irrational is not. Also, prove that there exists no function $g:\mathbb{R} \rightarrow \mathbb{R}$ such that any irrational is its period and any rational is not.

Solution

Sum of sets

In $\mathbb{R}$ we define the sum of sets as follows:

$$A+B= \{a+b, \; a \in A, \; b \in B \}$$

Find the sum $[a, b]+[c, d]$. 

Solution