MathDB

a) Prove that the function

F:\mathbb{R}\rightarrow \mathbb{R},\ F(x)=2\lfloor x\rfloor-\cos(3\pi\{x\})

is continuous over

\mathbb{R}

and for any

y\in \mathbb{R}

, the equation

F(x)=y

has exactly three solutions.
b) Let

k

a positive even integer. Prove that there is no function

f:\mathbb{R}\rightarrow \mathbb{R}

such that

f

is continuous over

\mathbb{R}

and that for any

y\in \text{Im}\ f

, the equation

f(x)=y

has exactly

k

solutions

(\text{Im}\ f=f(\mathbb{R}))

Problem Statement