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.

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.

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.

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.

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.

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

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.

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.

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.