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Continuity and Differentiability

ICSE · Class 12 · Mathematics

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

105 questions29 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

1

1. Limits and One-Sided Limits

  • A limit describes the real number that a function approaches as x approaches a point c.
  • A limit exists only when both the left-hand limit and the right-hand limit exist and are equal.
  • Right-hand limit means approaching c from the right side.
2

2. Continuity at a Point and on an Interval

  • A function is continuous at x=c when the limit at c equals the function value at c.
  • For continuity, both the limit and f(c) must exist and be equal.
  • A function is right continuous if the right-hand limit equals the function value.
3

3. Algebra of Continuous Functions

  • Sum, difference, and product of continuous functions are continuous.
  • A quotient of continuous functions is continuous if the denominator is non-zero.
  • 1/f is continuous if f(c)≠0.
4

4. Differentiability at a Point and One-Sided Derivatives

  • Differentiability at x=c means the derivative limit exists finitely.
  • The derivative exists only when both one-sided derivatives exist finitely and are equal.
  • Right differentiable and left differentiable are one-sided versions of differentiability.

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Full Notes

Key Concepts

A function approaches a real numberThe rightA function is continuous at xA function is discontinuous if f(c)A removable discontinuity occurs when

Frequently Asked Questions

What are the important topics in Continuity and Differentiability for ICSE Class 12 Mathematics?
Key topics in Continuity and Differentiability include Decision Tree: Is a Function Continuous at Point c?, Continuity and Differentiability - Complete Overview, Flowchart showing how limits are determined by comparing left and right limits. These are the concepts ICSE Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Continuity and Differentiability — ICSE Class 12 Mathematics?
Understand the core concepts first, then work through the 105 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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