Continuity and Differentiability

Board: CBSE |Class:Class12

ComprehensivestudynotesforContinuityandDifferentiabilitybyAjayYadav.

KeyConcepts

Continuity

fiscontinuousatx=aiflim(x→a)f(x)=f(a).fiscontinuouson[a,b]ifcontinuousateverypoint.Polynomials,trig,exponential,logarecontinuousontheirdomains.

Differentiability

f'(a)=lim(h→0)[f(a+h)-f(a)]/h.Differentiable⇒continuous.Continuous≠⇒differentiable.Example:|x|atx=0.

ChainRule

dy/dx=(dy/du)(du/dx).Forcompositefunctionsf(g(x)):f'(g(x))g'(x).

ImplicitDifferentiation

Forf(x,y)=0:differentiatebothsidesw.r.t.x,treatingyasy(x),andsolvefordy/dx.

LogarithmicDifferentiation

Fory=[f(x)]^eorproduct/quotientwithpowers:takelnbothsides,differentiate.

Rolle'sTheorem

Iffiscontinuouson[a,b],differentiableon(a,b),andf(a)=f(b),then∃c&in(a,b)withf'(c)=0.

Lagrange'sMeanValueTheorem

Iffcontinuouson[a,b],differentiableon(a,b),then∃c&in(a,b):f'(c)=[f(b)-f(a)]/(b-a).

ImportantFormulas

Continuity	lim(x→a)f(x)=f(a)
Chainrule	dy/dx=(dy/du)(du/dx)
Rolle’s	f(a)=f(b)⇒f'(c)=0forsomec
LMVT	f'(c)=[f(b)-f(a)]/(b-a)

SolvedExamples

Example1:Checkcontinuityoff(x)=|x-2| at x=2.

Solution: LHL=RHL=f(2)=0. Continuous. Differentiable? LHD=-1, RHD=1. Not differentiable.

Example 2: Find dy/dx if x²+y²=25 implicitly.

Solution: 2x+2y(dy/dx)=0 ⇒ dy/dx = -x/y.

Example 3: Find dy/dx for y = xx using logarithmic differentiation.

Solution: ln y = x ln x. (1/y)dy/dx = ln x + 1. dy/dx = xx(ln x+1).

Practice Questions

1. Check continuity of f(x)=sinx/x at x=0.
2. Find dy/dx if xy + y² = 1.
3. Differentiate y = (sinx)^e.
4. Verify Rolle’s for f(x)=x²-4 on [-2,2].
5. Verify LMVT for f(x)=x² on [1,3].

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