已知 $f(x)=\left\{\begin{array}{ll}x^2 & 0 \leq x < 1 \\ 1 & 1 \leq x \leq 2\end{array}\right.$ ，设
$$
F(x)=\int_1^x f(t) \mathrm{d} t(0 \leq x \leq 2) ，
$$
则 $f(x)$ 为
A. $\left\{\begin{array}{l}\frac{1}{3} x^3, 0 \leq x < 1 \\ x, 1 \leq x \leq 2\end{array}\right.$
B. $\left\{\begin{array}{l}\frac{1}{3} x^3-\frac{1}{3}, 0 \leq x < 1 \\ x, 1 \leq x \leq 2\end{array}\right.$
C. $\left\{\begin{array}{l}\frac{1}{3} x^3, 0 \leq x < 1 \\ x-1,1 \leq x \leq 2\end{array}\right.$
D. $\begin{cases}\frac{1}{3} x^3-\frac{1}{3} & 0 \leq x < 1 \\ x-1 & 1 \leq x \leq 2\end{cases}$