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.