We are given that the function must satisfy the following conditions
We can rewrite
by property (i) and we know that
by property (ii).
Property (ii) can also be interpreted as "if the function undergoes a horizontal stretch by a factor of 1/a, then the function is not the same as if it were vertically stretched by a factor of a"
It should be noted also that additivity implies homogeneity for an rational constant a and for continuous functions. Also note that if
for some non-zero constant, then
and
, a contradiction. so
Let
f(x) = 0 if x=0
-1 if x<0
1 if x>0
f(x+0) = f(x) = f(x) + 0 = f(x) + f(0)
For x and y with the same sign, then f(x+y) = f(x) + f(y)
Without loss of generality if x>y and x is positive and y is negative and x not equal to y, x+y > 0 and f(x+y) = 1 f(x)+f(y) = 1-1 = 0
So we can't use a simple piecewise function...
I'm stumped for now but tell me what you have so far. I'll sleep on it.
EDIT: The irony is that I almost forgot about a certain concept called "Freshman's Dream". I just don't know how to formally write it down. Maybe this doesn't make any sense.
Let f(x) = x^2 but f(x+y) = x^2 + y^2 (in other words, it's foiling x+y but it has no middle terms) = f(x) + f(y)
f(ax) = a^2*x^2 = a^2f(x)