Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that $$f ( x + 1 ) = \frac { 1 } { 2 } f ( x ) \text { for all } x \in \mathbb { R } ,$$ and let $a _ { n } = \int _ { 0 } ^ { n } f ( x ) d x$ for all integers $n \geq 1$. Then: (A) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $\int _ { 0 } ^ { 1 } f ( x ) d x$. (B) $\lim _ { n \rightarrow \infty } a _ { n }$ does not exist. (C) $\lim _ { n \rightarrow \infty } a _ { n }$ exists if and only if $\left| \int _ { 0 } ^ { 1 } f ( x ) d x \right| < 1$. (D) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $2 \int _ { 0 } ^ { 1 } f ( x ) d x$.
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that
$$f ( x + 1 ) = \frac { 1 } { 2 } f ( x ) \text { for all } x \in \mathbb { R } ,$$
and let $a _ { n } = \int _ { 0 } ^ { n } f ( x ) d x$ for all integers $n \geq 1$. Then:\\
(A) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $\int _ { 0 } ^ { 1 } f ( x ) d x$.\\
(B) $\lim _ { n \rightarrow \infty } a _ { n }$ does not exist.\\
(C) $\lim _ { n \rightarrow \infty } a _ { n }$ exists if and only if $\left| \int _ { 0 } ^ { 1 } f ( x ) d x \right| < 1$.\\
(D) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $2 \int _ { 0 } ^ { 1 } f ( x ) d x$.