Continuity and Differentability

Continuity:

  • Definition: A function f(x) is continuous at a point c if three conditions hold:

    1. f(c) is defined.
    2. The limit of f(x) as x approaches c exists.
    3. The limit of f(x) as x approaches c equals f(c).
  • Types of Discontinuities:

    • Removable Discontinuity: A discontinuity that can be filled by redefining the function at that point.
    • Jump Discontinuity: The function "jumps" from one value to another at a specific point.
    • Infinite Discontinuity: When the function's value becomes infinite at a certain point.
    • Essential Discontinuity: A discontinuity that cannot be classified as removable, jump, or infinite.
  • Continuity of Composite Functions:

    • The composition of continuous functions is continuous.
    • f(g(x)) is continuous at x=c if both f(x) and g(x) are continuous at x=c.

Differentiability:

  • Definition: A function f(x) is differentiable at a point c if the derivative f(c) exists.

    • The derivative represents the rate of change of a function at a given point.
  • Conditions for Differentiability:

    • Existence of Derivative: The limit limh0f(c+h)f(c)h exists.
    • Continuity at the Point: If a function is differentiable at c, it must also be continuous at c.
  • Differentiability Rules:

    • Sum/Difference Rule: (f±g)(x)=f(x)±g(x)
    • Product Rule: (fg)(x)=f(x)g(x)+f(x)g(x)
    • Quotient Rule: (fg)(x)=f(x)g(x)f(x)g(x)(g(x))2
    • Chain Rule: (f(g(x)))=f(g(x))g(x)
  • Differentiability and Continuity:

    • Differentiability implies continuity, but continuity does not necessarily imply differentiability.
    • A function can be continuous at a point where it is not differentiable, such as at sharp corners or cusps.

Key Concepts:

  • Local Linearity: Differentiability implies that the function can be approximated by a straight line at that point.
  • Critical Points: Points where the derivative is zero or undefined; they can indicate maxima, minima, or points of inflection.

Testing Continuity and Differentiability:

  • Using Limits: Calculate limits to check continuity and differentiability at a point.
  • Graphical Analysis: Observing the graph to identify jumps, holes, or sharp turns to determine continuity and differentiability.