Consider the function
f(x)={x2,2x−1,x≤1,x>1.
(a) Analyze the behavior of f at x=1 in terms of continuity.
(b) Compute the limit of the difference quotient hf(1+h)−f(1) as h→0− and as h→0+. Show every algebraic step.
(c) Based on the results in part (b), decide whether f has a derivative at x=1. Provide a clear verbal explanation connecting your limit calculations to the formal definition of the derivative.
a. Part 1: Part (a) — Continuity analysis at x=1
b. Part 2: Part (b) — Left-hand and right-hand difference quotients
c. Part 3: Part (c) — Conclusion on differentiability with justification