Question

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?

Answer 1

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.