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Continuity and Differentiability

Punjab Board · Class 12 · Mathematics

Quick revision notes for Continuity and Differentiability — Punjab Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

44 questions25 flashcards5 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 Chapter 5: Continuity and Differentiability – Complete Overview, Decision Tree for Choosing Differentiation Method, Continuity and Differentiability – Complete Chapter Overview. These are the concepts Punjab Board Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Continuity and Differentiability — Punjab Board Class 12 Mathematics?
Understand the core concepts first, then work through the 44 practice questions available for this chapter. Revise formulas and 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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