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).