If , where denotes the greatest integer function,…

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

Problem

f(x) = \begin{cases} \lfloor 1 - 4x^2 \rfloor, & 0 \le x < 1 \\ \lfloor x^2 - 2x \rfloor, & 1 \le x < 2 \end{cases}

, where

\lfloor\ \rfloor

denotes the greatest integer function, then (1)

f(x)

is continuous for all

x \in [0, 2]

(2)

f(x)

is differentiable for all

x \in [0, 2] - \{1\}

(3)

f(x)

is differentiable for all

x \in [0, 2] - \left\{\frac{1}{2}, 1\right\}

(4) None of these