Class 12 Maths - WEST-BENGAL
Continuity and Differentiability
The chapter 'Continuity and Differentiability' is a fundamental pillar of Calculus in the Class 12 West Bengal Council of Higher Secondary Education (WBCHSE) Mathematics curriculum. It bridges the gap between limits and calculus applications like derivatives and integrals. Students learn to determine whether a function is unbroken (continuous) at a point and whether it has a smooth, well-defined tangent (differentiable). Mastery of this chapter is crucial for scoring high marks in board exams, as it forms the base for solving complex problems in tangent-normals, maxima-minima, and mean value theorems, often appearing in both short and long answer sections.
Start Learning FreeKey Concepts
Continuity at a Point
A function f(x) is continuous at x = a if the left-hand limit, right-hand limit, and the value of the function at that point are all equal.
Differentiability at a Point
A function is differentiable at x = a if the left-hand derivative and right-hand derivative at that point exist and are finite, meaning the graph has no sharp corners.
Relationship between Differentiability and Continuity
If a function is differentiable at a point, it is necessarily continuous at that point, but the converse is not always true.
Chain Rule
A formula for computing the derivative of a composite function, stating that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
Logarithmic Differentiation
A technique using logarithms to simplify functions of the form y = [f(x)]^[g(x)] or complex products before taking the derivative.
Important Formulas
Board Exam Info
In the West Bengal (WBCHSE) Class 12 Mathematics board examination, this chapter typically carries around 8 to 12 marks. Questions usually include 1-mark multiple choice questions (MCQs), 2-mark very short answer questions on checking continuity or differentiability at a given point, and 4-mark or 5-mark long answer questions involving logarithmic differentiation, parametric differentiation, or finding unknown constants making a piecewise function continuous.
Frequently Asked Questions
Are all continuous functions differentiable?
No. While every differentiable function is continuous, a continuous function can have sharp corners (like f(x) = |x| at x = 0) where it is not differentiable.
How do I find unknown constants (like k) for a function to be continuous?
Set the left-hand limit equal to the right-hand limit and equal to the value of the function at the given point, then solve the resulting algebraic equation for the unknown.
When should I use logarithmic differentiation?
You must use logarithmic differentiation when the function is a variable raised to another variable power, such as y = x^sin(x), or when there is a complex product and quotient of many functions.
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