Class 12 Mathematics Chapter 5: Continuity and Differentiability
CBSE 2026–27 | NCERT-aligned free resource
1. Continuity
A function f is continuous at x=a when limx→af(x)=f(a). Intuitively, the graph has no break at that point.
LHL = RHL = f(a) → continuous at a
2. Algebra of Continuous Functions
Under suitable domain conditions, sums, differences, products and quotients of continuous functions remain continuous where the quotient denominator is non-zero.
3. Differentiability
A function is differentiable at a if the derivative exists there. Differentiability implies continuity, but continuity does not necessarily imply differentiability.
Continuous ⇏ Differentiable
Example: f(x)=|x| is continuous at 0 but not differentiable at 0.
4. Derivative Rules
Use standard derivatives together with the sum, product, quotient and chain rules. For composite y=f(g(x)), dy/dx=f′(g(x))g′(x).
5. Parametric Differentiation
If x and y are functions of a parameter t, then dy/dx=(dy/dt)/(dx/dt), provided dx/dt≠0.
6. Logarithmic Differentiation
For complicated products or powers, take logarithms first. For y=[f(x)]^{g(x)}, logarithmic differentiation can simplify the derivative.
7. Rolle’s and Mean Value Theorems
Rolle’s theorem guarantees at least one point where f′(c)=0 when the required continuity, differentiability and equal-endpoint conditions hold. Lagrange’s Mean Value Theorem gives f′(c)=(f(b)−f(a))/(b−a).
Worked Example
For f(x)=x², the derivative at x=a is 2a. Geometrically, this is the slope of the tangent to the curve at that point.
Exam Traps
- Continuity alone does not guarantee differentiability.
- Check the hypotheses before applying Rolle’s theorem or LMVT.
- For piecewise functions, compare left and right limits and the function value.
Practice
- Test continuity of a piecewise function at a specified point.
- Differentiate composite functions using the chain rule.
- Use logarithmic differentiation for a variable power.
- Apply Rolle’s theorem after verifying its conditions.
- Apply LMVT to a suitable polynomial or continuous differentiable function.
Deep-Dive Revision & Worked Examples
Continuity
A function f is continuous at x=a when f(a) exists, lim(x→a)f(x) exists, and lim(x→a)f(x)=f(a).
Worked Example: Finding a Parameter
For a piecewise function, calculate the left-hand and right-hand limits at the joining point. Equating both to the function value gives the required parameter(s).
Differentiability
Differentiability at a point requires the left and right derivatives to exist and be equal. Differentiability implies continuity, but continuity alone does not guarantee differentiability.
Continuous at a ⇏ necessarily differentiable at a
Derivative Applications
Use logarithmic differentiation for products and powers such as y=xˣ. Taking logs gives ln y=x ln x; differentiating yields y′/y=ln x+1.
Higher-Order Derivatives
The second derivative f″(x) measures the rate of change of the first derivative and is useful in studying concavity and optimisation.
Practice
- Check continuity of a piecewise function at its joining point.
- Find an unknown constant so a function is continuous.
- Check differentiability from left and right derivatives.
- Differentiate implicit functions.
- Use logarithmic differentiation for variable powers.
- Find first and second derivatives of a composite function.
Common Mistakes — Chapter 5
- Checking only one-sided limits at a joining point.
- Assuming continuity automatically means differentiability.
- Forgetting to evaluate the function value when testing continuity.