POLYA028:  A functional equation (proposed by Stephen C. Locke, 3/1/02).

Suppose that f is a real-valued function defined on real numbers, and that f(f(y))=f(y) and that f(x+y)=f(x)+f(y). Describe all such functions.

This is a simple problem, approximately suitable for students in the first rigorous real analysis course. The problem is built on a problem from one of the journals (Two Year College Journal? one of the recent issues).

Discussion. [LZ, 3/4/02]: The reference is: Problem719, College Mathematics Journal, Jan.(2002). The problem states:
If f is a real-valued function defined on real numbers, then f(x+y+f(y))=f(x)+2f(y) iff f(f(y))=f(y) and f(x+y)=f(x)+f(y).
I just found a proof of it. The deadline for submitting solutions to CMJ719 is 4/15/02, so maybe we should be careful not giving away the solution too much in our
discussion here.
 

Bibliography.

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