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Continuity and Differentiability — Revision Notes

Punjab Board · Class 12 · Mathematics

Continuity and Differentiability revision notes for Punjab Board Class 12 Mathematics: 4 topics in quick points.

44 questions25 flashcards8 formulas & key relations5 concepts

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A graph of a continuous function at a specific point, showing the limit approaching the function value, illustrating that the function is unbroken and defined at that point.
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Key Topics to Revise

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1. Continuity of a Function

  • A function f(x) is continuous at x = c if: lim(x→c) f(x) = f(c). This single condition contains three sub-conditions: (a) f(c) must exist, (b) lim(x→c) f(x) must exist, and (c) both must be equal.
  • For the limit to exist, the Left Hand Limit (LHL) must equal the Right Hand Limit (RHL): lim(x→c⁻) f(x) = lim(x→c⁺) f(x).
  • A function is continuous on an interval if it is continuous at every point in that interval.
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2. Algebra of Continuous Functions

  • If f and g are both continuous at x = c, then: f+g, f-g, f·g are all continuous at x = c.
  • f/g is continuous at x = c provided g(c) ≠ 0.
  • A constant multiple λ·f is continuous wherever f is continuous.
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3. Differentiability

  • The derivative of f at c is: f'(c) = lim(h→0) [f(c+h) - f(c)] / h, provided this limit exists.
  • f is differentiable at c if both the Left Hand Derivative (LHD) and Right Hand Derivative (RHD) exist and are equal.
  • LHD at c = lim(h→0⁻) [f(c+h) - f(c)] / h
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4. Chain Rule and Derivatives of Composite Functions

  • Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx).
  • In words: differentiate the outer function keeping inner function unchanged, then multiply by derivative of inner function.
  • For three levels: if y = f(v), v = g(u), u = h(x), then dy/dx = (dy/dv)·(dv/du)·(du/dx).

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Full Notes

Key Concepts

A function f is continuous atA function is continuous onPoints where a function failsIf f and g are continuousA function f is differentiable at

Frequently Asked Questions

What are the important topics in Continuity and Differentiability for Punjab Board Class 12 Mathematics?
Key topics in Continuity and Differentiability include Continuity of a Function, Algebra of Continuous Functions, Differentiability, Chain Rule and Derivatives of Composite Functions. Study these first, then practise questions on each for the Punjab Board Class 12 board exam.
How should I revise Continuity and Differentiability for the Punjab Board Class 12 board exam?
Learn the core ideas first, then work through the 44 practice questions on Continuity and Differentiability. Revise definitions regularly and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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All Punjab Board Class 12 Mathematics Revision Notes

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