Okay, this may end up being a rather mundane question, but I wanted to make sure I wasn’t missing something obvious.
There’s a problem from my analysis text that is either a misprint, or a new use of terminology, or I’ve read it completely wrong. As it is, I don’t think the problem is correct.
All typographical symbols are as they appear in the book (oo = infinity).
Problem: Suppose that f’ exists and is increasing on (0, oo) and that f is continuous on [0, oo) with f(0) = 0. Show that g(x) = [f(x)]/x is increasing on (0,oo).
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Now the only use I’ve seen for the operator in this book (aside from order of operations) is to indicate the greatest integer function. But that means that g(x) isn’t increasing, since for each piece of the function, it’s actually decreasing. What makes this even more suspect is that the hint says, “Show that g’(x) > 0.” So is there another use of that might work here?
Unless I’m wrong and the problem can be solved, my guess is that maybe g(x) is just f(x)/x (which I think works out, but I haven’t fully proven).
Or this is a new use of ‘increasing’ (despite an earlier example apparently explicitly contradicting this definition) to mean that there exists for every x1 in g’s domain, an x2>x1 such that g(x2)>g(x1). Prior to this ‘increasing’ seemed to only mean ‘strictly monotone increasing’; which is the accepted meaning?
In a completely unrelated problem, I’m also trying to work this one out :
Show that f(x) is differentiable, but the derivative is not continuous at x=0.
f(x) = x[sup]2[/sup]sin(1/x), x != 0
. . . .0, x= 0