Continuity and Differentiability
This domain tests the definition of continuity at a point, the link between continuity and differentiability, and the standard derivative techniques: chain rule, implicit differentiation, logarithmic differentiation and derivatives of inverse trigonometric functions. Rolle's and Lagrange's Mean Value Theorems also appear as 3–4 mark questions.
Key facts
- f is continuous at a if lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a); all three must match.
- Differentiability at a implies continuity at a, but continuity does not imply differentiability (consider |x| at 0).
- Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).
- d/dx (sin⁻¹x) = 1/√(1 − x²), d/dx (tan⁻¹x) = 1/(1 + x²).
- Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f′(c) = 0 for some c in (a, b).
Worked example
Test f(x) = |x| at x = 0. The left limit is 0 and the right limit is 0, and f(0) = 0, so f is continuous. For the derivative, the left quotient gives (0 − h − 0)/(−h) = −1, while the right quotient gives (h − 0)/h = +1. The one-sided derivatives differ, so f is not differentiable at 0.
Traps
- Checking only the limit and not the value f(a) at the join point.
- Forgetting to split the function at the point where its definition changes.
- Applying the power rule to x^x; use logarithmic differentiation instead.
- Differentiating sin⁻¹x without the square-root factor.