Multivariable Calculus · free preview

Continuity in Several Variables

The definition of continuity for functions of two and three variables, the algebra and composition rules that build continuous functions from simple pieces, and why limits and continuity diverge at broken points.

1. Continuity means the limit lands on the value

Continuity means the limit lands on the value

2. Removable breaks versus permanent breaks

Removable breaks versus permanent breaks

3. Algebra and composition assemble continuity

Algebra and composition assemble continuity

4. Continuity is the license for substitution

Continuity is the license for substitution

5. Classifying a break at the origin

The function f(x,y)=x2−y2x2+y2f(x, y) = \dfrac{x^2 - y^2}{x^2 + y^2} has no value at the origin. Can f(0,0)f(0,0) be defined so that ff becomes continuous there?

6. A composition and its continuity

Why is f(x,y)=sin⁡(x2+y2)f(x, y) = \sin(x^2 + y^2) continuous at every point of the plane?

7. Quotients and their domains

At which points is f(x,y)=xyx−yf(x, y) = \dfrac{xy}{x - y} continuous?

8. A removable break repaired

The function g(x,y)=x3−y3x2+y2g(x, y) = \dfrac{x^3 - y^3}{x^2 + y^2} has a removable break at the origin. What value should be assigned to g(0,0)g(0,0) to make gg continuous there, as a plain number?

9. Substitution through continuity

Compute lim⁡(x,y)→(1,2)xy+x2x+y\lim_{(x,y) \to (1,2)} \dfrac{xy + x^2}{x + y} and give the result as a plain number or fraction like 3/2.

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