The function is often quoted as an example of a function that is continuous…
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
The function g(x)=∣x∣ is often quoted as an example of a function that is continuous everywhere but not differentiable at x=0.
(i) Establish the continuity of g at x=0 using the definition of continuity.
(ii) Evaluate the limits of the difference quotient hg(0+h)−g(0) as h→0− and h→0+. What do these limits show?
(iii) Explain, in words, why the existence of different one–sided derivatives at a point prevents the function from being differentiable there, even though it may still be continuous.