MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C785BA.77DEA640" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C785BA.77DEA640 Content-Location: file:///C:/753AC878/2007prob8.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Problem 8

Problem 8<= /b>

 

Prove that if a real valued function f satisfies the condition f(f(x))=3Dx+1, then for any integer n,= f(n) is not an integer.

 

[Problem submitted by Steve Lee, LACC Professor of Mathematics.]

 

 

 

Sol= ution for Problem 8:

 

f(f(f(x))) =3D f(x+1), f(f(f(x))) =3D f(x)+1, therefore = f(x+1) =3D f(x) + 1……….(1)

 

Replace x by 0, 1, 2, 3… in equation (1), we get <= o:p>

f(1) =3D f(0) + 1,

f(2) =3D f(1) + 1 =3D f(0) + 1 += 1 =3D f(0) + 2,

f(3) =3D f(2) + 1 =3D f(0) + 2 += 1 =3D f(0) + 3

 

f(n) =3D f(0) + n……= ;…………………………R= 30;……………………(2)

 

Assume f(0) =3D k, where k is= an integer.

Then by equation (2),

f(k) =3D f(0) + k  f(f(0)) =3D= f(0) + f(0) 0+1=3D f(0)+ f(0) f(0)=3D1/2, a contradiction. Therefore f(0) can not an integer.

 

Then by equation (2), f(n) is= not an integer.

 

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