Lecture 9

Nov 11, 2024

Mathematics: Calculus - Lecture 9: Continuous Functions

Overview

  • University: Université libre de Bruxelles, Solvay Brussels School of Economics and Management
  • Author: Germain Van Bever
  • Key Topics:
    • Continuity
    • Stability results
    • Intermediate value theorem
    • Continuous function on a compact set

Continuity

  • Definition:
    • Function ( f ) is continuous at ( a ) if ( \lim_{x \to a} f(x) = f(a) ).
    • Requires ( a \in Df ) and ( a \in adh(Df \setminus {a}) ).

Illustrations

  • Illustration 1:
    • Function ( f(x) = \frac{((x+1)^2 - 1)^2}{x^2} ).
    • ( a = 0 ) not in domain, so continuity at ( 0 ) cannot be studied.
  • Illustration 2:
    • ( f(x) = \begin{cases} \frac{((x+1)^2 - 1)^2}{x^2} & \text{if } x \neq 0 \ 0 & \text{if } x = 0 \end{cases} )
    • Not continuous at ( 0 ) since ( \lim_{x \to 0} f(x) = 4 \neq f(0) ).
  • Illustration 3:
    • Function is continuous at ( 0 ) since ( \lim_{x \to 0} f(x) = f(0) = 4 ).

Continuity to the Right/Left

  • Definition:
    • Continuous to the right at ( a ): ( \lim_{x \to a^+} f(x) = f(a) )
    • Continuous to the left at ( a ): ( \lim_{x \to a^-} f(x) = f(a) )
  • Illustrations:
    • Function may not be continuous to both sides, affecting overall continuity.

Continuity of ( f )

  • Definition 9.3:
    • ( f ) is continuous if for all ( a \in Df ), ( f ) is continuous at ( a ).
    • ( f ) is continuous on ( A \subseteq Df ) if ( f|_A ) is continuous.

Lipschitz Continuity

  • Definition 9.4:
    • ( f ) is Lipschitz if there exists ( C > 0 ) such that ( \frac{|f(x_1) - f(x_2)|}{|x_1 - x_2|} \leq C ) for all ( x_1, x_2 \in I ).
  • Theorem 9.5:
    • A Lipschitz function is continuous on its interval ( I ).

Continuity and Standard Operations

  • Theorem 9.6: If ( f ) and ( g ) are continuous at ( a ):
    • ( f+g ) is continuous
    • ( cf ) is continuous
    • ( fg ) is continuous
    • ( f/g ) is continuous if ( g(a) \neq 0 ).

Continuity and Composition

  • Theorem 9.7: If ( f ) is continuous at ( a ) and ( g ) is continuous at ( f(a) ), then ( g \circ f ) is continuous at ( a ).

Continuity of Standard Functions

  • Functions such as ( x^r ), ( \exp(x) ), ( \ln x ), ( \sin x ), etc., are continuous on their domains.

Intermediate Value Theorem

  • Theorem 9.9: A continuous function on a convex set maps it to a convex set. For any ( w ) between ( u ) and ( v ), there exists an ( s ) such that ( f(s) = w ).

Fixed Point Theorem

  • Theorem 9.10: For a continuous function ( f ) mapping ([a, b]) to itself, a point ( c ) exists where ( f(c) = c ).

Continuous Function on a Compact Set

  • Theorem 9.11: The image of a compact set under a continuous function is compact.
  • Boundary Theorem (9.12): A continuous function on a compact set touches the boundaries of its image set.

Additional Insights

  • Continuous functions may not map bounded sets to bounded sets or closed sets to closed sets.
  • Exam questions explore the domain of continuity and asymptotes.

This lecture provides foundational insights into the behavior and properties of continuous functions, emphasizing their stability and implications in mathematical analysis.