Calculus Chapter on Continuous Functions

What condition must be met for a function f to be continuous at the point (a, b)?

Which of the following combinations of continuous functions will also yield a continuous function?

Under what condition is the composite function g(f(x, y)) continuous at (a, b)?

Which of the following statements about rational functions is true?

Which of the following functions is continuous?

Continuity of Functions

A function ( f: D \to R ) is continuous at point ( (a, b) ) in ( D ) if for every ( \epsilon > 0 ), there exists ( \delta > 0 ) such that if ( |(x, y) - (a, b)| < \delta ), then ( |f(x, y) - f(a, b)| < \epsilon ).
A function is continuous on D if it is continuous at each point ( (a, b) ) in the domain ( D ).

Algebra of Continuous Functions

If ( f ) and ( g ) are continuous at ( (a, b) ), then the following are also continuous:
- The sum ( f + g )
- The scalar multiple ( rf ) where ( r \in R )
- The product ( fg )
- The quotient ( 1/f ) if ( f(a, b) \neq 0 )

Composition of Continuous Functions

For functions ( f: D \to E ) and ( g: E \to R ):
- If ( f ) is continuous at ( (a, b) ) and ( g ) is continuous at ( f(a, b) ), then the composite function ( g \circ f ) is continuous at ( (a, b) ).

Examples of Continuous Functions

Polynomials: Functions like ( p(x, y) = x^2 + y^2 ) and ( p(x, y) = 2x^3y - 3x + y + 1 ) are continuous.
Rational Functions: A rational function ( r(x, y) = p(x, y)/q(x, y) ) is continuous at ( (a, b) ) if ( q(a, b) \neq 0 ).
Other Functions: Functions such as ( f(x, y) = x^3 \sin |y| + \cos(x^2 + y) ) and ( f(x, y) = e^{x^2 + xy} ) are also continuous.

This quiz explores the definitions and properties of continuous functions in calculus, specifically focusing on the epsilon-delta definition. Students will examine how continuous functions behave under various operations, such as addition and multiplication. Prepare to test your understanding of these fundamental concepts!