가우스 함수로 정의된 함수의 성질

ㄱ. $\displaystyle \frac{4}{3} = 1.333 \cdots$ 이므로 $0 \leq k \leq 66$ ; $\displaystyle \left[ x + \frac{k}{100} \right] = 1$, $67 \leq k \leq 99$ ; $\displaystyle \left[ x + \frac{k}{100} \right] = 2$ 이므로 $\displaystyle f \left( \frac{4}{3} \right)$$= 67 \times 1 + 33 \times 2$$= 133$ ㄴ. $n = 2 m$ (짝수) 이면 $\displaystyle \left[ x + \frac{n}{2} \right] =$$\displaystyle \left[ x + m + \frac{k}{100} \right] = m + \left[ x + \frac{k}{100} \right]$이므로 $\displaystyle f \left( x + \frac{n}{2} \right) = 100 m + f ( x ) = 50 n + f ( x )$ $\cdots \text{①}$ $n = 2 m - 1$ (홀수) 이면 $\displaystyle \left[ x + \frac{n}{2} \right] =$$\displaystyle \left[ x + m - \frac{1}{2} + \frac{k}{100} \right]$ 의 값은 $0 \leq k \leq 49$ ; $\displaystyle m - 1 + \left[ x + \frac{k + 50}{100} \right]$ $50 \leq k \leq 99$ ; $\displaystyle m + \left[ x + \frac{k - 50}{100} \right]$ 이므로 $\displaystyle f \left( x + \frac{n}{2} \right) = 50 ( m - 1 ) + 50 m + f ( x )$$= 50 n + f ( x )$ $\cdots \text{ ②}$ $\text{ ①} , \text{ ②}$에서 $\therefore$$\displaystyle f \left( x + \frac{n}{2} \right) = f ( x ) + 50 n$ (참) ㄷ. $\displaystyle \frac{n}{100} \leq x < \frac{n + 1}{100}$ 일때, $f ( x ) = n$ 이므로 $f ( f ( x ) - 1 ) = f ( n - 1 ) = 100 n - 100$ $nf ( x ) - 1 = n ^{2} - 1$ 식에 대입하면 $100 n - 100 = n ^{2} - 1$ $n ^{2} - 100 n + 99 = 0$ $n = 1 , 99$ $\therefore$2개 (거짓)