Class 12 Continuity and Differentiability Notes: Concepts, Formulas, Tests and Examples

Continuity and differentiability questions in Class 12 often become challenging when you have to work with piecewise functions, modulus expressions, limits or functions defined at specific points. Applying the wrong condition or confusing LHL and RHL can lead to errors, even when the underlying calculation is simple.

A clear understanding of continuity conditions, differentiability tests and derivative rules helps you approach these problems systematically. PW provides Class 12 learning through live lectures, recorded lectures, subject-wise DPPs and tests.

What Is Continuity in Class 12 Maths?

A function is continuous at a point if its limit at that point exists and equals the actual value of the function. Graphically, continuity means that the function has no break at the point being considered.

A function may be continuous on one interval but discontinuous on another. Therefore, its domain and the point under examination must always be considered.

Examples:

• sin x is continuous for every real number.

• |x| is continuous everywhere, including at x = 0.

• tan x is discontinuous at points where it is undefined, such as x = (2n + 1)π/2, where n is an integer.

Continuity at a Point: LHL, RHL and Function Value

A function f(x) is continuous at x = a when its left-hand limit, right-hand limit and actual function value are equal.

The mathematical condition is:

LHL = RHL = f(a)

That is:

lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = f(a)

Here:

• Left-Hand Limit (LHL): The value approached by f(x) as x approaches a from the left.

• Right-Hand Limit (RHL): The value approached by f(x) as x approaches a from the right.

• Function Value: The actual value of the function at x = a.

If the LHL and RHL are unequal, the limit does not exist at that point, so the function cannot be continuous there. If both limits exist and are equal but differ from f(a), the function is also discontinuous.

Memory Tip: Minus indicates the left side, and plus indicates the right side. Always compare both limits with the actual function value.

How to Check Continuity of a Piecewise Function?

Piecewise functions use different expressions over different intervals. To check continuity at the point where the definition changes, calculate the LHL, RHL and function value separately.

Follow these steps:

1. Use the expression valid for x < a to calculate the LHL.

2. Use the expression valid for x > a to calculate the RHL.

3. Find f(a) using the expression defined at x = a.

4. Compare all three values.

5. Conclude whether the function is continuous or discontinuous.

Example:

Consider the piecewise function:

f(x) = 2 − x, x < 2

f(x) = 2 + x, x ≥ 2

Check continuity at x = 2.

Step 1: Calculate LHL

Using the expression 2 − x:

LHL = 2 − 2 = 0

Step 2: Calculate RHL

Using the expression 2 + x:

RHL = 2 + 2 = 4

Step 3: Calculate the function value

f(2) = 2 + 2 = 4

Since:

LHL ≠ RHL

The function is discontinuous at x = 2.

Finding Unknown Constants Using Continuity

Continuity conditions are frequently used to find unknown constants in piecewise functions.

If a function contains constants and is continuous at x = a, apply:

LHL = RHL = f(a)

Example:

Suppose the values obtained at x = 1 are:

• LHL = 5a − 2b

• RHL = 3a + b

• f(1) = 11

Using the continuity condition:

5a − 2b = 3a + b = 11

This gives two equations:

3a + b = 11

2a = 3b

From the second equation:

a = 3b/2

Substituting into the first:

3 × (3b/2) + b = 11

11b/2 = 11

b = 2, a = 3

Therefore:

a = 3, b = 2

If a constant does not appear in the continuity equations, the continuity condition at that point may not be sufficient to determine its value.

Important Results and Properties of Continuous Functions

Several standard functions are continuous throughout their respective domains. These properties help simplify continuity questions.

The following functions are continuous wherever they are defined:

• Polynomial functions

• Sine and cosine functions

• Absolute value functions

• Exponential functions

• Logarithmic functions within their domains

• Sums and differences of continuous functions

• Products of continuous functions

• Scalar multiples of continuous functions

• Quotients of continuous functions wherever the denominator is non-zero

For a function of the form |g(x)|, if g(x) is continuous, then |g(x)| is also continuous wherever it is defined.

Examples:

• |sin x| is continuous for all real x.

• |x² + 2x| is continuous for all real x.

• |tan x| is continuous wherever tan x is defined.

Continuity of Special Functions

Some functions require special attention because their behaviour changes at particular points.

Greatest Integer Function

The greatest integer function, represented by [x], gives the greatest integer less than or equal to x.

Examples:

[1.9] = 1, [3.2] = 3, [3] = 3

The greatest integer function is:

• Continuous at every non-integer.

• Discontinuous at every integer.

At an integer n:

LHL = n − 1

RHL = n

f(n) = n

Since the LHL and RHL are unequal, the function is discontinuous at every integer.

Functions Involving sin(1/x)

Consider:

f(x) = x² sin(1/x), x ≠ 0

Since:

−1 ≤ sin(1/x) ≤ 1

Multiplying by x², we get:

−x² ≤ x² sin(1/x) ≤ x²

As x → 0, both outer expressions approach zero. Therefore, by the Squeeze Theorem:

lim (x→0) x² sin(1/x) = 0

If f(0) = 0, the function is continuous at x = 0.

In contrast, consider:

f(x) = 1/x

As x approaches zero:

• From the left, 1/x → −∞.

• From the right, 1/x → +∞.

Thus, the limit does not exist as a finite value at zero, and the function is discontinuous there.

What Is Differentiability in Class 12 Maths?

Differentiability measures the instantaneous rate of change of a function. A function is differentiable at a point if its derivative exists at that point.

The derivative of f(x) at x = a is defined as:

f′(a) = lim (h→0) [f(a + h) − f(a)] / h

This limit represents the derivative at the given point, provided it exists.

For a function to be differentiable at x = a, its left-hand derivative and right-hand derivative must be equal.

LHD = RHD

That is:

lim (h→0) [f(a − h) − f(a)] / (−h) = lim (h→0) [f(a + h) − f(a)] / h

Here:

• Left-Hand Derivative (LHD): Derivative calculated as x approaches a from the left.

• Right-Hand Derivative (RHD): Derivative calculated as x approaches a from the right.

If both derivatives are finite and equal, the function is differentiable at that point. If they are unequal, the function is not differentiable there.

How to Check Differentiability of Modulus Functions?

Modulus functions often contain sharp corners, so their differentiability must be checked carefully.

For a modulus expression, first remove the modulus according to the sign of the expression inside it.

Consider:

f(x) = |x − 1|

Its piecewise form is:

f(x) = 1 − x, x < 1

f(x) = x − 1, x ≥ 1

At x = 1:

For x < 1:

LHD = −1

For x > 1:

RHD = 1

Since:

LHD ≠ RHD

The function is not differentiable at x = 1, although it is continuous there.

For expressions of the form |g(x)|, start by solving:

g(x) = 0

These points are candidates for checking differentiability. However, not every zero necessarily creates a sharp corner, so the LHD and RHD must still be compared.

Examples:

• |2x − 1| is not differentiable at x = 1/2.

• |sin x| is not differentiable at x = nπ, where n is an integer.

• |x| + |x − 1| is not differentiable at x = 0 and x = 1.

Memory Tip: For |g(x)|, first solve g(x) = 0. Then check the left-hand and right-hand derivatives at those points.

Relationship Between Continuity and Differentiability

Continuity and differentiability are closely related, but they are not the same mathematical property.

The key theorem is:

Differentiability ⇒ Continuity

Every function differentiable at a point is also continuous at that point. However, the reverse is not always true.

A function may be continuous but not differentiable at a sharp corner, cusp or point where its left-hand and right-hand derivatives are unequal.

For example, f(x) = |x| is continuous at x = 0, but it is not differentiable there because its LHD and RHD are different.

Therefore, differentiable functions form a subset of continuous functions.

Basic Differentiation Rules and Formulas

Differentiation rules help solve derivatives of polynomial, trigonometric, exponential and logarithmic functions. These formulas are also used when differentiating composite functions.

Standard Derivative Formulas

Product Rule

The product rule is used when a function is the product of two differentiable functions.

If:

y = uv

Then:

dy/dx = u · (dv/dx) + v · (du/dx)

Quotient Rule

The quotient rule is used when one differentiable function is divided by another.

If:

y = u/v

Then:

dy/dx = [v · (du/dx) − u · (dv/dx)] / v²

Chain Rule

The chain rule is used to differentiate composite functions, where one function is contained inside another.

If:

y = f(g(x))

Then:

dy/dx = f′(g(x)) · g′(x)

Example:

Differentiate:

y = (sin(log x))³

Applying the chain rule:

dy/dx = 3 (sin(log x))² · cos(log x) · (1/x)

Therefore:

dy/dx = 3 sin²(log x) cos(log x) / x

Memory Tip: For composite functions, differentiate the outer function first and multiply it by the derivative of the inner function.

Continuity and differentiability form the foundation of Class 12 calculus. Understanding the conditions involving LHL, RHL and function values helps solve continuity questions, while comparing LHD and RHD determines differentiability. Special functions, modulus expressions and standard derivative rules become easier to handle once these concepts are clear.

Regular practice with different types of questions is important for improving accuracy in calculus. PW supports Class 12 exam preparation through live lectures, recorded lectures, Maths DPPs and tests. These resources provide opportunities to revise concepts, practise questions and assess progress throughout the course.