가우스·사인 연속

ㄱ. $f ( 0 ) g ( 0 ) = 0 \times 0 = 0$ $\displaystyle \lim\limits _{x \rightarrow 0} {f ( x ) g ( x ) = {\begin{cases} \lim\limits _{x \rightarrow 0 -} {f ( x ) g ( x ) = ( - 1 ) \times 0 = 0} & \\ \lim\limits _{x \rightarrow 0 +} {f ( x ) g ( x ) = 0 \times 0 = 0} & \end{cases}}}$ $\therefore x = 0$에서 연속 ㄴ. $\displaystyle {\begin{cases} f ( g ( 0 ) ) = f ( 0 ) = 0 & \\ \lim\limits _{x \rightarrow 0} {f ( g ( x ) ) = {\begin{cases} \lim\limits _{x \rightarrow 0 -} {f ( g ( x ) ) = f ( - 0 ) = - 1} & \\ \lim\limits _{x \rightarrow 0 +} {f ( g ( x ) ) = f ( + 0 ) = 0} & \end{cases}}} & \end{cases}}$ $\therefore x = 0$에서 불연속 ㄷ. $g ( f ( 0 ) ) = g ( 0 ) = 0$ $\displaystyle \lim\limits _{x \rightarrow 0} {g ( f ( x ) ) = {\begin{cases} \lim\limits _{x \rightarrow 0 -} {\begin{aligned} g ( f ( x ) ) = g ( - 1 ) = 0 \\ \end{aligned}} & \\ \lim\limits _{x \rightarrow 0 +} {g ( f ( x ) ) = g ( 0 ) = 0} & \end{cases}}}$ $\therefore x = 0$에서 연속 따라서 옳은 것은 ㄱ, ㄷ이다.