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.