Class 12 Maths - MP
Continuity and Differentiability
Continuity and Differentiability is a fundamental calculus chapter in the Class 12 Mathematics curriculum prescribed by the Madhya Pradesh Board (MPBSE). It builds upon limits studied in Class 11. Continuity explores whether a function can be drawn without lifting the pen, determined by the equality of left-hand limit, right-hand limit, and the value of the function. Differentiability checks if a sharp corner exists at a point. This chapter carries significant weight in the MPBSE board exams, featuring heavily in both short-answer questions and long-answer problems, including theorems like Rolle's Theorem and Lagrange's Mean Value Theorem.
Start Learning FreeKey Concepts
Continuity at a Point
A function f(x) is continuous at x = c if the left-hand limit, right-hand limit, and the value of the function at c are all equal.
Differentiability at a Point
A function is differentiable at x = c if its left-hand derivative and right-hand derivative exist and are equal at that point.
Relationship between Continuity and Differentiability
Every differentiable function is continuous, but a continuous function is not necessarily differentiable (e.g., at a sharp corner).
Logarithmic Differentiation
A technique used to find derivatives of functions in the form of variable raised to the power variable, like f(x)^g(x), by taking logarithms on both sides.
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 c in (a,b) where f'(c) = 0.
Important Formulas
Board Exam Info
In the Madhya Pradesh (MPBSE) Class 12 Mathematics board examination, the chapter 'Continuity and Differentiability' typically carries around 8 to 12 marks. Students can expect objective-type questions (MCQs/fill-in-the-blanks), 2-mark questions on checking continuity or finding simple derivatives, 4-mark questions on logarithmic differentiation or implicit functions, and occasional 4 or 5-mark questions involving derivative proofs or Mean Value Theorems.
Frequently Asked Questions
Are all continuous functions also 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 at x = 0 because of a sharp corner.
When should we use logarithmic differentiation?
Logarithmic differentiation is used when a function is given as the product or quotient of many functions, or when a function is in the variable-to-variable form, such as y = x^x.
How many steps are required to prove continuity at a point in board exams?
You need to show three things clearly: find the left-hand limit (LHL), right-hand limit (RHL), and the exact value of the function at that point, and then state that since LHL = RHL = f(c), the function is continuous.
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