Given that f (a+b) = f (a) + f (b) and f (x) is always positive, what os the value of f (0) ?

This is a strange question, and f (x) may not even exist. Why do I say that? Well ...

[1] We know that f (a+b) = f (a) + f (b). Therefore, f (0+0) = f (0) + f (0). In other words, f (0) = f (0) + f (0). Subtracting, we see, f (0) - f (0) = f (0) or 0 = f (0).

[2] So, what's the problem? We found the answer, f (0) = 0, right? Maybe, but the second rule says that f (x) is always positive. However, f (0) = 0 is not positive!

Since there is a contradiction, we must either conclude that the single value f (0) does not exist, or that the entire function f (x) does not exist.

To fix this, we could instead say that "f (x) is always nonnegative" and then we would be safe.