Hey kjtruitt.
f(x) = 3 is not a local maximum since there exists an b such that f(b) > f(3) or that there exists an x such that f(x) > f(3).
Consider the graph of f(x) = x for a domain of integers only. Could you pick the interval (1,6) and say that the point at (5,5) is a local maximum? My text states that
"f has a local maximum at the point (a,b) if and only if there is an open interval I containing a for which f(a) >= f(x) for all x in I."
Also seems given this definition that f(x) = 3 (or any constant) would be comprised of local maxima. And yet it seems clear that the intention is that a local maximum has points in the interval on each side of it.
Thanks for your reply, but I don't know if I understand it. Anyway you are responding to the second assertion in my post, so I'll talk about that second assertion as well. I was saying that if you have a horizontal line that passes through the y-axis at 3, then any interval of x values will all yield 3 as a result. So f(x) for any value in any interval, for any value in the domain of f(x)=3, is >= f(x) for all x in I. They are all equal, and so in that definition any x you might pick is a local maximum. Or so it seems to me. I just think the definition is a bit imprecise.
Now you said f(x)=3 is not a local maximum since there exists an b such that f(b) > f(3). But that's not the case with f(x)=3.
Another way to get at what I'm concerned about is to ask what is it about the definition of local maximum that describes a peak, with lesser values to each side, rather than just an incline with lesser values just to one side (ie a line of slope 1 in my example), or even a plateau with equal values to each side? Given the definition i posted it seems to be that the only thing that is disqualified (just lines here, don't know how to describe slope of curves yet) from local maximum are points on a negative sloped line. Get me?
Yes you are correct about local maximums being a peak with lower values on each side.
In terms of the neighborhood argument, it means that at the maximum there is an open interval where each side of the maximum is always less than the value of the maximum which is why you have your "peak" with each side being less.
In the case of the straight line there are no peaks which means we just have to look at the constraint for the domain.
If for example the line continued across all x in R, then the line would technically not have a proper maximum.