Asymptotic Properties of Functions in Complexity Analysis When dealing with the analysis of time and space complexity of algorithms, is it safe to assume that any function which has tight bounds ( i.e. $f(n)=\Theta(g(n))$ is asymptotically positive and asymptotically monotonically increasing.  I mean that for all $n$ greater than or equal to some $n_0$ both those properties hold?
It is safe to assume that $f$ is everywhere positive, hence, asymptotically positive. You can't use negative time or space.

It is not safe to assume that $f$ is asymptotically monotonically increasing, since this excludes constant functions, i.e., those which are $\Theta(1)$.

It isn't even safe to assume that $f$ is asymptotically monotonically non-decreasing; this precludes functions that oscillate. A good question might be "do any useful algorithms have asymptotically oscillating time or space complexities," but certainly you could create an algorithm that did.

I suppose a more rigorous answer would ask what your definition of "asymptotically monotonically increasing" means. If it means that it's $\Theta(g(n))$ where $g(n)$ is monotonically increasing for positive $n$, then the answer would be yes, by definition.