Right, i'm doing semester 2 analysis and i'm completely stumped, have no idea where to start on this question:
Let f be a map from an interval I to the reals (not necessarily continuous). Prove the equivalence of the two statements:
(i) if x,y are contained within I with f(x)<f(y), then for all c in (f(x),f(y)) there exists z in (x,y) union (y,x) : f(z) =c
(ii) if J, a subset of I, is any interval, then f(J) is an interval
any help would be appreciated, thanks
I'm not really sure about the last part you wrote in part (i) but lets see if I understand this:
I'll start you off with the easier direction:
Proof (ii) implies (i)
If J is an interval then f(J) is an interval,
so chose x, y in I, f(x) < f(y) and look at the interval [x,y] (or [y,x])
By assumption, f([x,y]) is an interval that obviously contains (if not is equal to) the interval [f(x), f(y)]. Thus, if c is in (f(x), f(y)), then c is in f([x,y]), and so there must exists a z in (x,y) such that f(z) = c.