Chapter 5: Continuity and Differentiability

Jul 23, 2024

Chapter 5: Continuity and Differentiability

Introduction

  • Importance: Continuity and Differentiability are key concepts in Class 12 Mathematics, forming the foundation for understanding Differentiation and Integration.
  • Focus: The lecture covers Exercise 5.1, which mainly deals with the concept of continuity.

Continuity

Definition

  • Meaning: Continuity refers to something that is unbroken or uninterrupted. In Mathematics, it means a function that has no breaks, jumps, or gaps in its graph.
  • Continuous Graph: A graph where you don't need to lift your pen while drawing.
  • Discontinuous Graph: A graph that has a break or gap where you would have to lift your pen.

Graphical Representation

  • Example: sin(x): The graph is continuous, with no breaks.
  • Example: f(x) = x²: The graph is continuous, smoothly extending in both directions.
  • Example: f(x) = |x|: The graph is continuous, forming a V-shape without breaks.
  • Discontinuous Example: A function that changes its value at a point, causing a break in the graph.

Checking Continuity Using Limits

  • Approach: Use limits to determine if a function is continuous at a point.
  • Concepts:
    • Limit x → c f(x): Value function approaches as x approaches but does not reach c.
    • Check limits from both sides (left and right).
    • If limit from the left = limit from the right = function value at that point, then the function is continuous.
  • Example: For f(x) with different definitions on different intervals, check limits from left and right of the point.

Definition of Continuous Function Using Limits

  • For a function f(x) to be continuous at x = c:
    • f(c) should be defined
    • lim (x → c) f(x) should exist
    • lim (x → c) f(x) = f(c)
  • Discontinuous: If any of the above conditions fail.

Algebra of Continuous Functions

Operations

  1. Addition: If f(x) and g(x) are continuous, then f(x) + g(x) is continuous.
  2. Subtraction: If f(x) and g(x) are continuous, then f(x) - g(x) is continuous.
  3. Multiplication: If f(x) and g(x) are continuous, then f(x) * g(x) is continuous.
  4. Division: If f(x) and g(x) are continuous, then f(x)/g(x) is continuous provided g(x) ≠ 0.

Composite Functions

  • Definition: Composite functions like fog and gof involve combining two continuous functions.
  • If f(x) and g(x) are continuous, then fog(x) and gof(x) are also continuous.

Example Problems

Example 1: Polynomial Function

  • Given: f(x) = 5x − 3
  • Check: Continuity at x = 0, x = −3, and x = 5.
  • Solution:
    • Check continuity by computing f(c) and lim (x → c) f(x) for each point.
    • Prove that f(c) = lim (x → c) f(x) for each x = 0, x = −3, and x = 5.
    • Conclude the function is continuous at these points.

Key Takeaway

  • Polynomial Functions: Always continuous because their graphs never break.

Conclusion

  • The lecture provides foundational understanding of continuity, how to check it using limits, and the behavior of continuous functions under algebraic operations.
  • Further understanding can be achieved through continuous practice and solving different types of problems.

Reminder: Any polynomial function will be continuous at any point because its graph is unbroken.

  • Action: Review limits and practice more problems from Exercise 5.1.
  • Next Steps: Continue to Exercise 5.2 and further differentiation topics.