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Questions and Answers
What condition must be met for a function f to be continuous at the point (a, b)?
What condition must be met for a function f to be continuous at the point (a, b)?
- For every ϵ > 0, there is δ > 0 such that ∥(x, y) − (a, b)∥ < δ implies |f(x, y) − f(a, b)| < ϵ. (correct)
- For every δ > 0, there is ϵ > 0 such that ∥(x, y) − (a, b)∥ < ϵ.
- For every δ > 0, there is ϵ > 0 such that |f(x, y) − f(a, b)| < δ.
- If f is defined at (a, b), then f is continuous.
Which of the following combinations of continuous functions will also yield a continuous function?
Which of the following combinations of continuous functions will also yield a continuous function?
- All polynomial functions defined on the same domain.
- The reciprocal of a continuous function if the function is non-zero at the point. (correct)
- Only the sum of two continuous functions.
- The product of two continuous functions. (correct)
Under what condition is the composite function g(f(x, y)) continuous at (a, b)?
Under what condition is the composite function g(f(x, y)) continuous at (a, b)?
- If g is continuous at f(a, b) regardless of f's continuity.
- If g(f(x, y)) is a polynomial function.
- If both f is continuous at (a, b) and g is continuous at f(a, b). (correct)
- If f is continuous at (a, b) and g is not continuous at f(a, b).
Which of the following statements about rational functions is true?
Which of the following statements about rational functions is true?
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Which of the following functions is continuous?
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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