Continuity and Differentiability
Continuity and Differentiability tests the limit-based definitions everything later depends on: continuity at a point through left-hand and right-hand limits, differentiability through the derivative limit, and derivative rules extended to composite, implicit, inverse-trigonometric, exponential and parametric forms. Rolle's Theorem and the Mean Value Theorem are also examined here.
Definitions and rules
- f is continuous at a if lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a). All three must exist and agree.
- Differentiable at a implies continuous at a. The converse fails: f(x) = |x| is continuous but not differentiable 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), d/dx(aˣ) = aˣ log a, d/dx(log x) = 1/x.
- Parametric form: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.
- Rolle's Theorem needs continuity on [a, b], differentiability on (a, b) and f(a) = f(b). The MVT needs the first two and concludes f′(c) = (f(b) − f(a))/(b − a).
Worked example
Verify Rolle's Theorem for f(x) = x² − 4 on [−2, 2]. As a polynomial, f is continuous on [−2, 2] and differentiable on (−2, 2), and f(−2) = 0 = f(2). Now f′(x) = 2x = 0 gives c = 0, and 0 lies in (−2, 2). The conclusion holds.
Traps
- Applying Rolle's Theorem or the MVT without verifying every hypothesis.
- Finding f′(c) = 0 but not confirming c lies in the open interval.
- Testing only f(a) when the question is about continuity of a piecewise function at a.
- Differentiating xˣ as x·x^(x−1); logarithmic differentiation is required.