A function f is defined analytically by
f(x)={∣x−2∣,(x−2)2,x≤2,x>2.
Answer the following questions about the behavior of f at x=2.
(1) Part 1: (a) Calculate limx→2−f(x) and limx→2+f(x). Determine whether f is continuous at x=2, and justify your conclusion.
(2) Part 2: (b) Compute limh→0−hf(2+h)−f(2) and limh→0+hf(2+h)−f(2). Show all algebraic steps.
(3) Part 3: (c) Using the results from part (b), determine whether f is differentiable at x=2. If the derivative fails to exist, state whether the point is a corner or a cusp and provide a brief justification based on the one-sided derivatives.