You could always say the lower bound is zero or some very small function. That's pointless and is not better in any sense.
lower bound like you are talking about (the function always takes longer than this to run for an input of size n)
lower bound across all inputs of size n (the function never runs faster than this, given any possible input of size n)
The two are different
EDIT: adding a link to the wikipedia article defining Big-O and Big-Omega: http://en.wikipedia.org/wiki/Big_O_notation
>It is not contradictory however, to say that the worst-case
>running time of insertion sort is [Big Omega](n^2),
>since there exists an input that causes the algorithm
>to take [Big Omega](n^2) time.
>Introduction to Algorithms, 2nd Edition, p 46