Thanks for the detail - self taught so I'll defer and try to take on board (also written quickly - I'm sure doing no help to my annoyance that sparked the response - the single minded and to me complex approach of most of these explanations - which I'm sure you'll agree many of which make the same mistakes I have)
one question though -
do you have a source for this 'irregularities' bit? I've never seen that mentioned before at all and a Google hasn't helped. doesn't seem to fit with 'upper bound' and 'worst case'.
[edit: also still not sure how you say im confusing for theta. that's what 'the largest number of times it might' was for.]
If you look at the Wikipedia article on big-O notation, it says that f(x) = O(g(x)) if and only if there exists some M > 0 and x_0 such that |f(x)| < M|g(x)| for all x > x_0. The x_0 is what lets you ignore early irregularities; without it, you couldn't say that sin(x) + 1 is O(x) since there's obviously no M such that sin(0) + 1 < M * 0.
one question though - do you have a source for this 'irregularities' bit? I've never seen that mentioned before at all and a Google hasn't helped. doesn't seem to fit with 'upper bound' and 'worst case'.
[edit: also still not sure how you say im confusing for theta. that's what 'the largest number of times it might' was for.]