Is this proposition true?
For all functions f
f(a+b) = f(a) + f(b).
If yes then why? If no then what are those special functions called and what property do they have?
EDIT: Wow i think floor/celing functions do not hold the property.? I can think of counterexamples but could somebody please prove this. But what are the functions that hold this property called?
In procedural programming this is not true, since arbitrary functions can have arbitrary side-effects.
For example, you could have a global counter that is incremented every time you call f() and added to the return value.
No.
They are called linear if also f(k * a) = k * f(a).
In general such a function is called a homomorphism, but this is not limited to addition.
You mention a function f for which:
f(a + b) = f(a) + f(b)
Such a function is called a Homomorphism, and it can be defined on certain algebraic structures. In this case + is a special binary function that maps a and b onto an element of the same domain.
Obviously not all functions are homomorphisms, as others have already shown you.