2008-10-30 23:16| 
38 Comments | 本文内容遵从首先呢，让我们来一个牛B函数大回顾。这下我不知道要赚多少的PV。你能否构造一个函数f(x)，使得：
它是一个阶梯状的连续函数？
它是除常函数之外的没有最小正周期的周期函数？
该函数只在一点连续？
该函数在[0,1]和(0,1)之间形成一一对应？
该函数某一点导数为正，但该点邻域不构成单增区间？
平面上任意小的圆内均包含函数上的点？
另外还有一些可能是众所周知（所以没在Blog里写过）的函数，比如处处连续但处处不可导的函数、在有理点处处不连续在无理点处处连续的函数等等。
好了，现在呢，又一个牛B东西出现了。你能不能想出这样一个函数f，它的定义域和值域都是R，并且对于任意小的区间l=(u,v)，这个函数都能把(u,v)满射到整个R上。换句话说，是否存在这样的函数f(x)，对于任意一个实数t以及任意一个区间(u,v)，总存在一个x满足u<x<v且f(x)=t。
首先，注意到函数h(x)=tan( pi*(x-1/2) )将(0,1)上的数与全体实数一一对应，因此，我们只需要找出一个g(x)，它能把任意小的区间满射到(0,1)区间上，然后在它上面复合一个h(x)就得到我们题目所求的函数了。
我们的g(x)可以这样定义：首先，把x写成三进制数。如果x的三进制表达里没有数字2或者有无穷多个数字2，那么g(x)=0。否则，x的三进制中含有有限多个数字2。找到最末的一个数字2，把它及其前面的部分都去掉，这样剩下的就只有0和1了。在这个01串前面加上一个小数点，把它当作一个二进制小数转换回去，作为g(x)的函数值。这样的话，g(x)就是一个值域为[0,1)的函数，并且在任意小的区间内，我们总能找到一个点，它所对的函数值就是你想要的那个值：只需要取一个该区间内的精度足够高（小数点位数足够多）的三进制有限小数，在后面加一个数字2，再后面的01串就可以想要啥就要啥了。
注意到g(x)把任意小的区间都映射到[0,1)上，而前面提到的h(x)则在(0,1)和R之间建立了一一对应的关系。于是，定义f(x)为，当g(x)=0时f(x)=0，否则f(x)=h( g(x) )。这就是满足题意的函数f。
题目来源：http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/Challenges/October2001.html
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楼上沙发牛
哈哈，这个题目的简化版做过，当时参加的是上海交大的自主招生考试，要求构造一个[0,1]对应(0,1)的函数（还有其他要求忘记了），好像当时我写的是sin(1/x)
唉...板凳没了
还是x*sin(1/x)+x这种形式的最漂亮。
Matrix大牛我想问你一个问题：
R和R X R的基是相等的，也就是说[0,1]区间和(x,y),0<=x<=1,0<=y<=1这个正方形区域上的点是一样多的，那么怎么构造它们之间的一个双射函数，我想了很久都没找到这样一个函数……
如果你知道的话，请指点一下.给我发封邮件吧，先谢过了^-^。
哇，您大牛可以研究一下选择公理了
他不是早研究过了吗……那个囚犯问题……
不少是数分作业题。。。
回复zmq：
把每个x写成小数0.x1x2...，每个y写成小数0.y1y2...，并让f(x,y)=0.x1y1x2y2...，这就是一个双射。
当然，要限制当可以用比如说0.9999...时不准用1.00000...之类的
路过。
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