Let I R be an interval and let f : I R be increasing on

Question:

Let I ⊂ R be an interval and let f : I → R be increasing on I. Suppose that c ∈ I is not an endpoint of I. Show that f is continuous at c if and only if there exists a sequence (xn) in I such that xn < c for n = 1, 3, 5, . . . ; xn > c for n = 2, 4, 6, . . . ; and such that c = lim(xn) and f(c) = lim (f(xn)).