Thread: Help with proving incresing function on an interval

A proper open subinterval of the open interval (a, b) subset of R is an open
interval (c, d) such that (c, d) is subset but not equal to (a, b) (i.e. a < c < d < b).

Suppose that f : (a, b) --> R is increasing on every proper open subinterval of
(a, b). Prove that f is increasing on (a, b).

Originally Posted by 450081592

A proper open subinterval of the open interval (a, b) subset of R is an open
interval (c, d) such that (c, d) not subset of (a, b) (i.e. a < c < d < b).

That "not" is incorrect. You mean to say (c,d) is a subset of (a,b)

Suppose that f : (a, b) --> R is increasing on every proper open subinterval of
(a, b). Prove that f is increasing on (a, b).

If you allowed (a,c) or (c, b) where a< c< b as "proper" subintervals, that would be easy.

If not, I think a proof by contradiction would be best. If f is not increasing on (a,b), there exist a< c< d< b with f(c)> f(d). Now look at the interval (c,d).

Originally Posted by HallsofIvy

That "not" is incorrect. You mean to say (c,d) is a subset of (a,b)


If you allowed (a,c) or (c, b) where a< c< b as "proper" subintervals, that would be easy.

If not, I think a proof by contradiction would be best. If f is not increasing on (a,b), there exist a< c< d< b with f(c)> f(d). Now look at the interval (c,d).

ya sorry I meant that (c,d) is subset but not equal to (a,b) i.e a proper subset