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
- 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.
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.