Step 1: Continuity Check For f(x) = {g(x), x<a; h(x), x>=a}: compute LHL = lim(x->a-) g(x), RHL = lim(x->a+) h(x), and f(a) = h(a). All must be equal.

Step 2: Differentiability Check Left derivative = lim(h->0-) [f(a+h)-f(a)]/h (use g's formula). Right derivative = lim(h->0+) [f(a+h)-f(a)]/h (use h's formula). Both must be equal and finite.

Step 3: Parameter Determination Continuity gives one equation. Equal derivatives gives another. Solve the system.

Step 4: Higher-Order Checks For $C^1$ (continuously differentiable): additionally verify that f' is continuous at a. This means lim f'(x) from both sides equals f'(a).

Common Pitfalls:

• Computing lim f'(x) instead of f'(a) via definition. These can differ ($x^2$ sin$\frac{1}{x}$ example).
• Forgetting that differentiability requires continuity first.
• Not checking both sides independently.

Example: f(x) = {$ax^{2+1}$, x<=1; 2x+b, x>1}. Continuity: a+1 = 2+b. Differentiability: 2a = 2, a=1. Then b=0.