Continuity and Differentiability
Learn Continuity and Differentiability with notes, examples, and practice questions.
Chapter Overview
You can draw a continuous curve without lifting your pen. Differentiable means no sharp corners.
A function is continuous at a point if its graph has no break, hole, or jump there. Continuity is the bridge between limits and differentiability.
Differentiable functions are always continuous — but continuous functions need not be differentiable (corners, cusps).
Continuity at a Point
Definition
f continuous at x=a iff lim(x→a) f(x) = f(a)
Three conditions
f(a) defined; limit exists; limit = f(a)
Types of Discontinuity
- Removable: limit exists but ≠ f(a) or f(a) undefined — fix by redefining one point
- Jump: left and right limits exist but unequal
- Infinite: function blows up at the point
Continuity on Intervals
- Polynomials, rational (where defined), trig, exponential, log — continuous on domain
- Sum, difference, product, quotient (denominator ≠ 0) of continuous functions — continuous
- Composition of continuous functions — continuous
- Intermediate Value Theorem: if f continuous on [a,b] and k between f(a) and f(b), ∃c with f(c)=k
Differentiability
Definition
f′(a) = lim(h→0) [f(a+h)−f(a)]/h exists
Relation
Differentiable at a ⇒ continuous at a
Not converse
|x| is continuous at 0 but not differentiable
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Check Continuity
Is f(x) = (x²−9)/(x−3) continuous at x = 3?
- 1f(3) undefined
- 2lim(x→3) = lim(x→3)(x+3) = 6
- 3Limit exists but f(3) not defined → removable discontinuity
Answer
Not continuous at x = 3 (removable)
Example 2: Piecewise Function
For f(x) = {x² if x≤1, 2x−1 if x>1}, check continuity at x=1.
- 1f(1) = 1
- 2Left limit = 1, right limit = 2(1)−1 = 1
- 3Limit = f(1) = 1 ✓
Answer
Continuous at x = 1
Example 3: |x| at 0
Is |x| differentiable at x = 0?
- 1Continuous at 0 ✓
- 2Left derivative = −1, right derivative = +1
- 3Unequal → not differentiable
Answer
Not differentiable at 0
Key Points to Remember
- ✓ Check LHL, RHL, and f(a) for piecewise functions
- ✓ Differentiability needs smooth tangent — no corner
- ✓ IVT proves root existence without finding exact value
Exam Tips
- • Always evaluate left and right limits separately at junction points
- • State which condition fails for discontinuity
- • Rolle's and Mean Value theorems need continuity on closed interval
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Continuity & Derivatives
- Continuous at a:lim(x→a⁻) f(x) = f(a) = lim(x→a⁺) f(x)
- Derivative:f′(a) = lim[h→0] [f(a+h) − f(a)]/h
- (xⁿ)′ = nxⁿ⁻¹
- (sin x)′ = cos x ; (cos x)′ = −sin x
- (tan x)′ = sec²x ; (cot x)′ = −cosec²x
- (sec x)′ = sec x tan x ; (cosec x)′ = −cosec x cot x
- (eˣ)′ = eˣ ; (aˣ)′ = aˣ ln a
- (log x)′ = 1/(x ln a) ; (ln x)′ = 1/x
Practice MCQs — Continuity and Differentiability
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
If f is differentiable at a, then f is: