Continuity and Differentiability
Continuity and Differentiability
This domain tests left and right continuity, differentiability at a point, derivatives of composite, implicit, inverse-trigonometric and exponential functions, and the mean value theorems.
Core rules
- f is continuous at a if lim(x→a) f(x) = f(a); for piecewise functions check LHL = RHL = value.
- Differentiability implies continuity, but continuity does not imply differentiability.
- Use the chain, product and quotient rules; learn d/dx(sin⁻¹x) = 1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²), d/dx(aˣ) = aˣ ln a.
- For a piecewise function, compare LHD and RHD at the join; equality gives differentiability.
- 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. Lagrange's MVT: f′(c) = [f(b)−f(a)]/(b−a).
Worked example
Test f(x) = |x| at x = 0.
Continuity: lim(x→0)|x| = 0 = f(0), so f is continuous. Left derivative: for h<0, (|h|−0)/h = −h/h = −1. Right derivative: for h>0, h/h = 1. Since −1 ≠ 1, f is not differentiable at 0. The corner in the graph confirms this.
Traps
- Assuming continuity at a point guarantees differentiability there.
- Missing the inner derivative in the chain rule, e.g. d/dx sin(x²) = 2x cos(x²).
- Writing d/dx(xˣ) as x·xˣ⁻¹; use logarithmic differentiation.
- Checking only one side of a piecewise join.
- Confusing the hypotheses of Rolle's theorem with its conclusion.