记$\left { x\right }=x-\left\lfloor x\right\rfloor\textgreater0$.  则$f(x)\leq 1\Leftrightarrow 2014^{\left {x\right }}-1\leq \frac{1}{\left\lfloor x\right\rfloor}$  $\Leftrightarrow \left {x\right }\leq\log_{2014}{\frac{\left\lfloor x\right\rfloor+1}{\left\lfloor x\right\rfloor}}$  于是单个满足$f(x)\leq 1$的区间长度为$\log_{2014}{\frac{\left\lfloor x\right\rfloor+1}{\left\lfloor x\right\rfloor}}$  区间总长度$=\sum_{x=1}^{2013}\log_{2014}{\frac{\left\lfloor x\right\rfloor+1}{\left\lfloor x\right\rfloor}}=\log_{2014}{\frac{2}{1}\times\frac{3}{2}\times\cdots\times\frac{2014}{2013}}=\log_{2014}{2014}=1$, 选$$\text{A}$$. 
2014AMC12A, 21  For every real number $$x$$, let $$\left\lfloor {x} \right\rfloor$$ denote the greatest integer not exceeding $$x$$, and let $$f(x)= \left\lfloor {x} \right\rfloor(2014^{x-\left\lfloor {x} \right\rfloor}- 1)$$. The set of all numbers $$x$$ such that $$1\leqslant x\textless{} 2014$$ and $$f(x)\leqslant1$$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?  对于任意实数$$x$$，令$$\left\lfloor {x} \right\rfloor$$ 表示不超过$$x$$ 的最大整数，令$$f(x)= \left\lfloor {x} \right\rfloor(2014^{x-\left\lfloor {x} \right\rfloor}- 1)$$. 满足 $$1\leqslant x\textless{} 2014$$ 和 $$f(x)\leqslant1$$ 的所有 $$x$$ 的集合是一些不相交区间的并集。 这些区间的长度之和是多少？

[1] $$1$$ 
[2] $$\frac{ \log 2015}{ \log 2014}$$ 
[3] $$\frac{ \log 2014}{ \log 2013}$$ 
[4] $$\frac{2014}{2013}$$ 
[5] $$2014^{\frac{1}{2014}}$$ 