Class 12 Maths - CBSE
Continuity and Differentiability
The chapter 'Continuity and Differentiability' builds the foundation of calculus in Class 12 CBSE Mathematics by connecting limits with graphs and rates of change. You will learn to determine if a function is continuous without gaps or jumps, and whether it is differentiable (smooth at every point). This chapter introduces advanced differentiation techniques like logarithmic differentiation, parametric form, and higher-order derivatives, heavily featuring the Chain Rule. It also covers major theoretical results like Rolle's Theorem and the Mean Value Theorem. Carrying significant weight in the CBSE board exam, mastering this chapter is essential for solving calculus applications like integrals.
Start Learning FreeKey Concepts
Continuity of a Function
A function is continuous at a point if its left-hand limit, right-hand limit, and the value of the function at that point are all equal.
Differentiability
A function is differentiable at a point if its left-hand derivative and right-hand derivative exist and are equal, meaning the graph has no sharp corners.
Relationship between Continuity and Differentiability
Every differentiable function is continuous, but a continuous function is not necessarily differentiable (e.g., modulus function at x = 0).
Logarithmic Differentiation
A technique used to differentiate functions of the form f(x)^g(x) or complex products/quotients by taking logarithms on both sides first.
Rolle's Theorem
If a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) where f'(c) = 0.
Important Formulas
Board Exam Info
In the CBSE Class 12 Mathematics exam, this chapter typically carries around 8 to 12 marks. Common question types include checking continuity and differentiability of piecewise functions at critical points, finding values of unknown constants (like k), complex chain rule problems, logarithmic differentiation, second-order derivatives, and proof-based questions on Rolle's Theorem or Mean Value Theorem.
Frequently Asked Questions
Are all continuous functions differentiable?
No. While every differentiable function is continuous, the converse is not true. For example, f(x) = |x| is continuous at x = 0, but it is not differentiable there due to a sharp corner.
When should I use logarithmic differentiation?
You should use logarithmic differentiation when a function is given in the form of a variable raised to another variable, i.e., f(x)^g(x), or when an expression involves a complicated product and quotient of many terms.
How do I prove a function is continuous at a specific point in board exams?
You must explicitly show three steps: evaluate the Left Hand Limit (LHL), Right Hand Limit (RHL), and the value of the function f(a). If LHL = RHL = f(a), the function is continuous.
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