If the range of the function is

MATHEMATICS · Subjective · AoPS · Mathematics · Special Functions · Special Functions

Problem

If the range of the function

f(x)=\left\{\frac{x}{4}\right\}+\cos\left(\frac{\pi(1-2[x])}{2}\right)+\sin\left(\frac{\pi[x]}{2}\right)

\left[\frac{\alpha}{4},\frac{\beta}{4}\right]\cup\left[\frac{\gamma}{4},\frac{\delta}{4}\right]\cup\left[\frac{2\gamma+1}{4},\frac{\delta}{2}\right]

, where

\{\,\cdot\,\}

and

[\,\cdot\,]

represent the fractional-part and greatest-integer functions respectively, find

\alpha^2+\beta^2+\gamma^2+\delta^2