One Variable Advanced Calculus

7.4. CONTINUOUS AND NOWHERE DIFFERENTIABLE 135If it did exist, thenHence, replacing h with −h,|h| − f ′ (0) h = o (h)|−h| − f ′ (0) (−h) = o (−h)and so, subtracting these,and so2f ′ (0) h = o (−h) − o (h) = o (h)2f ′ (0) = o (h)h .Now letting h → 0, it follows f ′ (0) = 0 if it exists. However, this would say|h| = o (h)which is false. Thus f ′ (0) cannot exist. However, this function has right derivatives atevery point and also left derivatives at every point. For example, consider f ′ (0) as aright derivative. For h > 0and so f ′ + (0) = 1. For h < 0,and sof (h) − 0 − 1h = 0 = o (h)f (h) = −hf (h) − f (0) − (−1) h = 0 = o (h)and so f ′ − (0) = −1. You should show f ′ (x) = 1 if x > 0 and f ′ (x) = −1 if x < 0.The following diagram shows how continuity at a point and differentiability thereare related.fis continuous at xf ′ (x)exists7.4 Continuous And Nowhere DifferentiableHow bad can it get in terms of a continuous function not having a derivative at somepoints? It turns out it can be the case the function is nowhere differentiable but everywherecontinuous. An example of such a pathological function different than the one Iam about to present was discovered by Weierstrass in 1872. Before showing this, hereis a simple observation.Lemma 7.4.1 Suppose f ′ (x) exists and let c be a number.f (cx) ,g ′ (x) = cf ′ (cx) .Then letting g (x) ≡