The function g(x)=∣x∣ is often quoted as an example of a function that is continuous everywhere but not differentiable at x=0.
a. (i) Establish the continuity of g at x=0 using the definition of continuity.
b. (ii) Evaluate the limits of the difference quotient hg(0+h)−g(0) as h→0− and h→0+. What do these limits show?
c. (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.