HMMT 二月 2011 · CALCCOMB 赛 · 第 8 题

英文解析

8. Let f : [0 , 1) → R be a function that satisfies the following condition: if
   ∞
   ∑
   x = = .a a a . . .na
   1 2 3
   10 nn =1
   is the decimal expansion of x and there does not exist a positive integer k such that a = 9 for allnn ≥ k , then
   ∞
   ∑
   f ( x ) = .na
   2 nn =110
   ( )
   ′1
   Determine f .
   ∑3
   ∞
   1 3
   Answer: 0 Note that = .
   n =1 n
   3 10
   Clearly f is an increasing function. Also for any integer n ≥ 1, we see from decimal expansions that
   1 1 1 1
   f ( ± ) − f ( ) = ± .
   n 2 n
   3 10 3 10 − n − 1 − n 1
   Consider h such that 10 ≤ | h | < 10 . The two properties of f outlined above show that | f ( +
   ∣ ∣3
   1 1
   ∣ f ( + h ) − f ( ) ∣
   1 1 1 1
   n +1
   3 3
   h ) − f ( ) | < . And from | | ≤ 10 , we get < . Taking n → ∞ gives h → 0
   ∣ ∣
   2 n n − 1
   3 10 h h 10
   ′ 1 1
   and f ( ) = lim = 0.
   n − 1
   n →∞
   3 10