Continuity and Differentiability
Complete Class XII Mathematics chapter notes covering limits, continuity, differentiability, derivatives, chain rule, implicit differentiation, logarithmic differentiation, parametric differentiation and second-order derivatives.
Introduction
Continuity and Differentiability is one of the fundamental chapters of Class XII Mathematics. The concepts of limit, continuity and derivative form the foundation of differential calculus.
Continuity describes whether a function behaves without a break at a point, while differentiability describes whether the function has a well-defined instantaneous rate of change or slope.
Limit
Describes the value approached by a function.
Continuity
No break or jump at the point.
Derivative
Measures instantaneous rate of change.
Limits
The limit of a function describes the value that the function approaches as the independent variable approaches a particular value.
Left-Hand Limit
Right-Hand Limit
Existence of Limit
The two-sided limit exists only when the left-hand limit and right-hand limit are equal and finite.
Continuity
A function f(x) is continuous at x = a if:
2. limx→af(x) exists
3. limx→af(x) = f(a)
Fundamental Condition
Continuity from Left and Right
Example
Check whether f(x)=x²+3x+2 is continuous at x = 1.
limx→1(x²+3x+2) = 1+3+2 = 6
Therefore, limx→1f(x)=f(1).
Hence the function is continuous at x = 1.
Properties of Continuity
If f and g are continuous at x = a, then the following functions are also continuous at x = a, wherever they are defined.
Sum of continuous functions
Difference of continuous functions
Product of continuous functions
Quotient when g(a) ≠ 0
Standard Continuous Functions
Polynomial, rational functions on their domains, exponential, logarithmic, trigonometric and inverse trigonometric functions are continuous wherever they are defined.
Differentiability
A function is differentiable at x = a if its derivative exists at x = a.
Left-Hand Derivative
Right-Hand Derivative
Differentiability Condition
Continuity and Differentiability
Differentiability
Continuity
If f is differentiable at a point, then f must be continuous at that point.
Continuity
Differentiability
A function may be continuous but not differentiable at a point.
Standard Example
The function f(x)=|x| is continuous at x = 0 but is not differentiable there because its left-hand and right-hand derivatives are different.
Derivatives
Important Standard Derivatives
Product Rule
Quotient Rule
Chain Rule
The chain rule is used when one function is composed with another function.
dy/dx = df/dg × dg/dx
Example
Find the derivative of:
y = sin u
dy/du = cos u
du/dx = 2x
Therefore,
dy/dx = cos u × 2x
= 2x cos(x²)
Implicit Differentiation
When x and y are related by an equation and y is not explicitly expressed as a function of x, differentiation is performed implicitly.
Important Rule
Whenever a term containing y is differentiated, multiply by dy/dx.
Example
2x + 2y dy/dx = 0
2y dy/dx = −2x
dy/dx = −x/y
Logarithmic Differentiation
Logarithmic differentiation is useful when a variable occurs in both the base and exponent, or when a function contains products and quotients that are difficult to differentiate directly.
Taking logarithm:
ln y = ln f(x)
Example
ln y = x ln x
Differentiating:
(1/y) dy/dx = ln x + 1
Therefore:
dy/dx = y(ln x + 1)
Since y = xˣ,
dy/dx = xˣ(ln x + 1)
Parametric Differentiation
If x and y are both expressed in terms of a parameter t, then:
Important Condition
The formula requires dx/dt ≠ 0 at the point under consideration.
Example
dy/dt = 3t²
Therefore,
dy/dx = (3t²)/(2t)
dy/dx = 3t/2 (t ≠ 0)
Second Order Derivatives
If y=f(x), then the first derivative is dy/dx. Differentiating it again gives the second derivative.
Second derivative: y″ = d²y/dx²
Example
Therefore,
y″ = 12x² − 6
Solved Examples
Example 1 — Check continuity of a polynomial
Polynomial functions are continuous for every real number.
Therefore f(x) is continuous for all x ∈ ℝ.
Example 2 — Find a parameter for continuity
ax + 1, x < 2
5, x = 2
x + a, x > 2
For continuity at x = 2:
RHL = 2 + a
f(2) = 5
Hence:
a = 2
The required value is a = 2.
Example 3 — Differentiate sin(x²)
dy/dx = cos(x²) × 2x
dy/dx = 2x cos(x²)
Example 4 — Product Rule
y′ = u v′ + v u′
= x² cos x + 2x sin x
Example 5 — Implicit differentiation
(x + 2y)dy/dx = −(2x+y)
dy/dx = −(2x+y)/(x+2y)
Example 6 — Logarithmic differentiation
(1/y)dy/dx = ln(x²+1) + x(2x)/(x²+1)
Therefore:
dy/dx = (x²+1)ˣ [ln(x²+1) + 2x²/(x²+1)]
Example 7 — Parametric differentiation
dy/dt = a cos t
Therefore:
dy/dx = (a cos t)/(−a sin t)
dy/dx = −cot t
Example 8 — Second derivative
y″ = 20x³ − 24x
Example 9 — Test differentiability of |x| at 0
RHD at x = 0 = 1
Since LHD ≠ RHD,
f(x)=|x| is not differentiable at x=0.
However, it is continuous at x=0.
Example 10 — Derivative of tan⁻¹x
Practice Questions
1. Find lim(x→2) (x²+3x+1).
2. Check whether f(x)=x²−4x+7 is continuous at x=2.
3. Find the derivative of x⁷.
4. Differentiate sin(3x).
5. Differentiate cos(x²).
6. Differentiate x²eˣ.
7. Differentiate ln(x²+1).
8. Find dy/dx if x²+y²=16.
9. Find dy/dx if x=t² and y=t³.
10. Find the second derivative of x⁴.
11. Is |x| differentiable at x=0?
12. State the relation between differentiability and continuity.
13. Differentiate tan⁻¹x.
14. Differentiate sec x.
15. Differentiate xˣ using logarithmic differentiation.
16. Find d²y/dx² for y=x³−5x.
17. Find the derivative of e^(2x+1).
18. Find dy/dx for x=a cos t, y=a sin t.
19. State the continuity condition at x=a.
20. State the differentiability condition at x=a.
HOT Questions
1. Find the value of k for which the function is continuous at x=1:
kx+2, x<1
5, x=1
x+k, x>1
Equate LHL, RHL and f(1).
2. Show that every differentiable function is continuous.
Start with the definition of derivative and write:
Taking h→0 proves that f(a+h)→f(a), hence continuity.
3. Find the value of a for which |x−a| is differentiable at x=a.
At x=a the graph has a corner. Compare the left and right derivatives.
The function is not differentiable at x=a.
4. If x=t+1/t and y=t−1/t, find dy/dx.
dy/dt = 1 + 1/t²
Therefore:
dy/dx = (1+1/t²)/(1−1/t²)
5. Find the derivative of y=(sin x)ˣ.
(1/y)y′ = ln(sin x)+x cot x
Therefore:
y′=(sin x)ˣ [ln(sin x)+x cot x]
6. Find d²y/dx² for y=sin x.
y″ = −sin x
7. Determine whether the function x|x| is differentiable at x=0.
For x≥0, x|x|=x² and for x<0, x|x|=−x².
Hence it is differentiable at x=0.
8. Find the condition for continuity of a piecewise function.
At every joining point, equate:
Quick Revision Sheet
| Concept | Important Formula / Result |
|---|---|
| Limit | lim(x→a) f(x) |
| Limit Exists | LHL = RHL |
| Continuity | lim(x→a)f(x)=f(a) |
| Differentiability | LHD = RHD |
| Derivative from First Principle | f′(a)=lim(h→0)[f(a+h)-f(a)]/h |
| Product Rule | (uv)′=uv′+vu′ |
| Quotient Rule | (u/v)′=(vu′−uv′)/v² |
| Chain Rule | dy/dx=(dy/du)(du/dx) |
| Implicit Differentiation | d(y)/dx = dy/dx |
| Parametric Differentiation | dy/dx=(dy/dt)/(dx/dt) |
| Second Derivative | d²y/dx² |
| Key Relation | Differentiability ⇒ Continuity |
Exam Tips
Check the point
Always identify the point where continuity or differentiability is being tested.
Continuity
Write LHL = RHL = f(a).
Differentiability
Compare LHD and RHD at piecewise or modulus functions.
Use Chain Rule
For composite functions, identify the inner and outer functions.
Implicit Functions
Remember to multiply differentiated y-terms by dy/dx.
Parametric Form
Use dy/dx = (dy/dt)/(dx/dt).
Logarithmic Differentiation
Take logarithms when the variable occurs in both base and exponent.
Show Working
Write intermediate steps clearly to avoid losing marks.
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