is discontinuous at x = 1. Explain how this function fails the continuous function guarantee.
If we try to get f(x) within 0.5 of f(1) = 2 (that is, 1.5 < f(x) < 2.5), we run into trouble. No matter how much we restrict x, since we aren't allowed to have 1 ≤ x ≤ 1, the picture will always have some values of x that are less than 1. Since f will be less than 1 for these values, f will not be between 1.5 and 2.5. We don't find the picture we want. Sniff.