If is a periodic function with period

MATHEMATICS · Single correct · AoPS · Mathematics · Special Functions · Special Functions

Problem

f(x) = \frac{\{x\}g(x)}{[x]g(x)}

is a periodic function with period

\frac{1}{4}

, where

g(x)

is a differentiable function, then (where

\{\cdot\}

denotes fractional part of

x

). (1)

g'(x)

has exactly three roots in

(\frac{1}{4}, \frac{5}{4})

(2)

g(x) = 0

x = \frac{k}{4}

, where

k \in \mathbb{I}

(3)

g(x)

must be non-zero function (4)

g(x)

must be periodic function