Exercise 18 solution: Suppose H is discontinuous at y_{0} ∈ R. Since H is increasing, there exist sequences {u_{i}} → y_{0} and {v_{i}} → y_{0} as i → ∞ satisfying u_{i} < u_{i+1}, v_{i+1} < v_{i} for all i, and

a = lim_{i→∞} H(u_{i}) < lim_{i→∞} H(v_{i}) = b.

Again using the fact H is increasing, we then have H(R) ∩ (a, b) = ∅. However, H(eo(x_{0})) = { iα + j : i, j ∈ Z } = eo(0, R_{α}) is dense in R by Exercise 8(b). This contradicts H(R) ∩ (a, b) = ∅; hence there is no such point y_{0}, and H is continuous on R.