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Questions and Answers
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?
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Which of the following functions is continuous?
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Study Notes
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.
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Description
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!
