합성함수 연속

$\displaystyle ( g \circ f ) ( 1 ) = g ( 1 ) = \frac{5}{2}$ $\displaystyle \lim\limits _{x \rightarrow 1 + 0} ( {g \circ f} ) ( x ) = g ( 1 ) = \frac{5}{2}$ $\displaystyle \lim\limits _{x \rightarrow 1 - 0} ( {g \circ f} ) ( x ) = g ( a ) = 2 ^{a} + 2 ^{-a}$ $( g \circ f ) ( x )$가 $x = 1$에서 연속이므로 $( g \circ f ) ( 1 )$$\displaystyle = \lim\limits _{x\rightarrow 1+0} ( g \circ f ) ( x ) = \lim\limits _{x\rightarrow 1-0} ( g \circ f ) ( x )$ $\displaystyle \therefore 2 ^{a} + 2 ^{-a} = \frac{5}{2}$ $2 \cdot 2 ^{2a} - 5 \cdot 2 ^{a} + 2 = 0$ $( 2 \cdot 2 ^{a} - 1 ) ( 2 ^{a} - 2 ) = 0$ $\displaystyle 2 ^{a} = \frac{1}{2} , 2$ $a = - 1 , 1$ ∴ 모든 실수 $a$의 곱 $= - 1$