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

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