Continuity and Differentiability - BSEB Inter Study Guide
Continuity and Differentiability
This section tests left and right continuity, differentiability at a join point, the chain rule, derivatives of inverse trigonometric functions and mean value theorems.
Core rules
- f is continuous at a if lim x→a⁻ f(x)=lim x→a⁺ f(x)=f(a).
- Differentiability implies continuity, but continuity does not imply differentiability: |x| at 0 is the standard counterexample.
- Chain rule: d/dx f(g(x))=f′(g(x))g′(x). Also d/dx sin⁻¹x=1/√(1−x²) for |x|<1.
- Rolle theorem needs continuity on [a,b], differentiability on (a,b), and f(a)=f(b); then f′(c)=0 for some c.
- Lagrange MVT: f′(c)=(f(b)−f(a))/(b−a).
Worked example
Verify MVT for f(x)=x² on [1,3] and find c. f is a polynomial, so continuous on [1,3] and differentiable on (1,3). (f(3)−f(1))/(3−1)=(9−1)/2=4. Set f′(c)=2c=4, so c=2, which lies in (1,3).
Traps
- Checking only f(a) and f(b) for continuity at a piecewise join.
- Assuming differentiability from a graph that has a corner.
- Using f′(c)=0 when the theorem only guarantees f′(c) equals the secant slope.
- Forgetting the interval condition |x|<1 for the inverse-sine derivative.