**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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