Lecture on Continuity and Differentiability

Introduction

• Welcome and Greetings: The lecture started with a warm welcome to the students. The lecturer explained the significance of Continuity and Differentiability in calculus.
• Engagement with Students: Emphasized the importance of interaction and preparation.

Continuity

Definitions and Concepts

• A function is continuous at a point if it can be drawn without lifting the pen.
• Types of Continuity:
  • Removable: Limits exist but the function is not defined at the point.
  • Non-Removable: Either the limits do not exist or are not equal.
    • Jump Discontinuity: Limits exist but are not equal.
    • Infinite Discontinuity: Limits tend to infinity.
    • Oscillatory Discontinuity: The function oscillates between values.

Mathematical Definition

• If a function is continuous at x = 1, then:
  • Left-hand limit (LHL)
  • Right-hand limit (RHL)
  • Function value at the point must be equal.
• Formula: [ ext{lim}{x o c^-} f(x) = ext{lim}{x o c^+} f(x) = f(c) ]

Differentiability

Definitions and Concepts

• Difference between Continuity and Differentiability.
• A function is differentiable at a point if it has a unique tangent at that point.
• Mathematical Definition:
  • If a function ( f ) is differentiable at ( x = a ): [ f'(x) = ext{lim}_{h o 0} rac{f(a+h) - f(a)}{h} ]
• Right-hand derivative (RHD) and Left-hand derivative (LHD) must be equal.
• Formula: [ ext{lim}{h o 0^+} rac{f(a+h) - f(a)} = ext{lim}{h o 0^-} rac{f(a-h) - f(a)}{h} ]_

Practical Examples

• Example showing a function that is continuous but not differentiable.

Graphical Interpretation

• Continuous function: Smooth curving graph, no breaks.
• Discontinuous Graphs:
  • Removable: A hole in the graph.
  • Jump: Sudden leap from one function value to another.
  • Infinite: Vertical asymptote.
  • Oscillatory: Rapid oscillation near a point.

Checking Continuity and Differentiability

• Procedure:
  1. Verify if the function is continuous over the range.
  2. Calculate LHD and RHD.
  3. If LHD = RHD, function is differentiable.
  4. If LHD ≠ RHD, function might be continuous but not differentiable.
  5. If limits involve infinity, further check discontinuity.

Algebra of Continuous and Differentiable Functions

Results

• The sum, difference, product of continuous functions is continuous.
• The same applies to differentiable functions.
• Result on Composition: If f and g are continuous, ( f(g(x)) ) is continuous.
• Non-Removable Discontinuity in Composition: Check must be manual.

Special Cases and Theorems

• Mean Value Theorem and Rolle's Theorem:
  • f is continuous on [a,b] and differentiable on (a,b)
  • There exists a c in (a,b) such that [ f'(c) = rac{f(b) - f(a)}{b - a} ]
• Application of Rolle's Theorem:
  • Ensure endpoints have the same function values.
  • Ensure the preconditions are met for application.

Decisional Scenarios for Application

• When not to use derivative directly: If LHD or RHD involve infinity or oscillations, check continuity separately.

Practice Examples and Questions

• Real-life applications of the discussed principles.
• Solving past IIT questions.
• Identifying points of discontinuity and checking differentiability.
• Converging on set results for quick problem-solving.

Conclusion

• Recap of key concepts from Continuity and Differentiability.
• Emphasis on practice and real-life applications.
• Encouragement for continuous preparation and study.

Final Notes

• Motivational Segment: Stories and quotes to inspire continuous learning and perseverance.
• Interactive Q&A: Addressing student questions and clarifications.
• Practical Assignments: Encouragement to solve additional problems for better understanding.

Have a great night and keep doing the hard work!

Motivational Shair (Poem) for Students

• If they had not broken your heart, you might not have gained the resolve to pursue your dreams.
• Read, learn, and even if you feel down, keep striving towards your success.
• Your teacher, who stood by you, gave you time and effort, reminding you: "Keep working hard, the sky is your limit!"