Now there are 4 case possible:

Case 1: a is even b is even

Case 2: a is even b is odd

Case 3: a is odd b is even

Case 4: a is odd b is odd

rewriting the f(a) and f(b) in each case:

Case 1: \(f(a)= a-1\) and \(f(b)= b-1\) so .... \(f(a)+f(b)= a+b-2\)

Case 2: \(f(a)= a-1\) and \(f(b)= b+1\) so .... \(f(a)+f(b)= a+b\)

Case 3: \(f(a)= a+1\) and \(f(b)= b-1\) so .... \(f(a)+f(b)= a+b\)

Case 4: \(f(a)= a+1\) and \(f(b)= b+1\) so .... \(f(a)+f(b)= a+b+2\)

\(f(a)+f(b)= a+b\) holds only when either a is odd and b is even or vice versa.

Hence option C is correct!
_________________

Sandy

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