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Problem 3

Let S S be the set of non-negative integers. Find all functions f:S→S f:S\to S such that f(m+f(n))=f(f(m))+f(n) f(m+f(n))=f(f(m))+f(n) for all m,n∈S m,n\in S .
Step 1 of 4: Find the initial identities
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In plain words

Find the initial identities

f(0)=0f(0)=0
Detailed analysis

Putting m=n=0 m=n=0 gives f(f(0))=f(f(0))+f(0) f(f(0))=f(f(0))+f(0), so f(0)=0 f(0)=0. Putting m=0 m=0 then gives f(f(n))=f(n) f(f(n))=f(n) for every n n , and the equation becomes f(m+f(n))=f(m)+f(n) f(m+f(n))=f(m)+f(n).