I take it you mean all functions that have property #1. Since a> 0 that simply means that there exist two distinct numbers, x and y, such that f(x)= f(y).

Let x and y be any two numbers. Then we can take a= |y-x| so that f(x)= f(x+a)= f(y).ii. Characterize all functions from 4.

There are many many easy examples. Try!iii. Find a function that is 1, but not 2,3,4

iv. Find a function that is 2, but not 3 and 4.

v. Find a function that is 3, but not 2 and 4.

Thanks!