Continuity and Differentiability

 

Continuity and Differentiability: A Brief Overview

Continuity

• Definition: A function is continuous at a point if its limit at that point exists and equals its value at that point.  
• Types:
  • Pointwise continuity: Continuous at a single point.  
  • Uniform continuity: Continuous across an entire interval.
• Properties:
  • Sum, difference, product, and quotient of continuous functions are continuous.
  • Composition of continuous functions is continuous.  
  • Intermediate Value Theorem: If a function is continuous on an interval and takes on two different values, then it must take on every value between those two values.  

Differentiability

• Definition: A function is differentiable at a point if its derivative exists at that point.  
• Geometric Interpretation: The derivative at a point represents the slope of the tangent line to the graph of the function at that point.  
• Relationship with Continuity: A differentiable function is always continuous, but the converse is not true (e.g., absolute value function at x=0).
• Properties:
  • Sum, difference, product, and quotient rules for derivatives.  
  • Chain rule for differentiating composite functions.  
  • Mean Value Theorem: If a function is continuous on a closed interval and differentiable on its interior, then there exists at least one point in the interval where the tangent line is parallel to the secant line connecting the endpoints.

Key Concepts and Theorems

• Intermediate Value Theorem
• Mean Value Theorem
• Rolle's Theorem (a special case of the Mean Value Theorem)  
• Taylor's Theorem (approximating functions with polynomials)

Applications

• Optimization problems (finding maximum or minimum values)
• Related rates (finding the rate of change of one variable with respect to another)  
• Physics (e.g., velocity and acceleration)
• Engineering (e.g., modeling physical systems)

Note: These are just brief summaries. For a deeper understanding, it's recommended to explore textbooks or online resources that provide more detailed explanations, examples, and exercises.