Continuity and Differentiability is one of the most concept-driven chapters in Class 12 Mathematics and acts as the backbone of calculus. The uploaded PDF is a well-structured, NCERT-aligned resource that divides the chapter into three clear parts: Continuity and Differentiability, Derivatives, and Rolle’s Theorem with Mean Value Theorem. It contains quick reviews, tips, tricks, multiple-choice questions, and solved short and long answer questions, making it useful for both concept building and exam revision.
I am writing about this topic because many students understand differentiation formulas but struggle when questions involve continuity, piecewise functions, or theoretical results like Rolle’s Theorem and MVT. This chapter demands clarity of logic more than memorisation. By presenting the key ideas from the PDF in a simple, practical, and organised way, I want to help learners strengthen their fundamentals and feel more confident while solving board and competitive exam questions.
Continuity of a Function at a Point
A function f(x) is said to be continuous at x = c if:
- lim x→c⁻ f(x) exists
- lim x→c⁺ f(x) exists
- lim x→c f(x) = f(c)
If any of these conditions fail, the function is discontinuous at that point.
The PDF clearly explains that polynomial, trigonometric, exponential, and logarithmic functions are continuous in their respective domains, while the greatest integer function is discontinuous at every integer.
Continuity in an Interval
- f(x) is continuous in (a, b) if it is continuous at every point in that interval
- f(x) is continuous in [a, b] if it is continuous in (a, b), right continuous at a, and left continuous at b
Geometrically, continuity means there is no break in the graph of the function.
Discontinuity and Composite Functions
A function is discontinuous at x = a if:
- LHL ≠ RHL
- LHL = RHL ≠ f(a)
- f(a) is not defined
If g is continuous at a and f is continuous at g(a), then the composite function (f ∘ g) is continuous at a.
Differentiability of a Function
A function f(x) is differentiable at x = c if:
lim h→0 [f(c + h) − f(c)] / h
exists.
The PDF emphasises:
- If a function is differentiable at a point, it is continuous there
- A function can be continuous but not differentiable
Example: f(x) = |x| is continuous at x = 0 but not differentiable at x = 0.
Left-Hand and Right-Hand Derivatives
- LHD = lim h→0⁻ [f(x + h) − f(x)] / h
- RHD = lim h→0⁺ [f(x + h) − f(x)] / h
For differentiability at a point, LHD must equal RHD.
Algebra of Derivatives
If f and g are differentiable functions, then:
- d/dx [f(x) ± g(x)] = f′(x) ± g′(x)
- d/dx [f(x)g(x)] = f(x)g′(x) + g(x)f′(x)
- d/dx [f(x)/g(x)] = [g(x)f′(x) − f(x)g′(x)] / [g(x)]²
These rules are widely used in the solved examples.
Download this CLASS 12 – CONTINUITY & DIFFERENTIABILITY PDF File: Click Here
Derivatives of Standard Functions
Some commonly used results:
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| eˣ | eˣ |
| log x | 1/x |
| sin⁻¹x | 1/√(1 − x²) |
Derivatives of Composite and Implicit Functions
Using chain rule:
dy/dx = (dy/du)(du/dx)
For implicit functions, both sides are differentiated with respect to x.
The PDF also covers logarithmic differentiation and parametric differentiation.
Rolle’s Theorem
If:
- f(x) is continuous on [a, b]
- f(x) is differentiable on (a, b)
- f(a) = f(b)
Then there exists at least one c in (a, b) such that:
f′(c) = 0
This theorem is often tested in theoretical and numerical questions.
Mean Value Theorem (MVT)
If:
- f(x) is continuous on [a, b]
- f(x) is differentiable on (a, b)
Then there exists c in (a, b) such that:
f′(c) = [f(b) − f(a)] / (b − a)
MVT helps in finding specific points where slope of tangent equals average rate of change.
Importance of This Chapter in Exams
- High weightage in CBSE boards
- Foundation for Applications of Derivatives
- Concept-based MCQs and case-study questions
Regular practice from NCERT examples and the PDF is essential.
