Let f(x) = 1 x and g(x) = 1 x if x > 0 8 + 1 x if x < 0 show that f '(x) = g'(x) for all x in their domains. can we conclude from the corollary below that f − g is constant? if f '(x) = g'(x) for all x in an interval (a, b), then f − g is constant on (a, b); that is, f(x) = g(x) + c where c is a constant. for x > 0, f(x) = g(x), so f '(x) = g'(x). for x < 0, f '(x) = and g'(x) = , so f '(x) g'(x). however, the domain of g(x) is not an interval [it is (−[infinity], 0) ∪ (0, [infinity])] so we cannot conclude that f − g is constant (in fact it is not).