The Calculus with Analytic Geometry

Louis Leithold

Chapter 19

Differential Calculus Of Functions Of Severai Variables - all with Video Answers

Educators

Section 3

Continuity Of Functions Of More Than One Variable

Problem 1

In Exercises 1 through 6 , discuss the continuity of $f$.
$f(x, y)=\left\{\begin{array}{ll}\frac{x y}{\sqrt{x^{2}+y^{2}}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right.$ (HINT: See Exercise 13, Exercises 19.2)

Donald Yeh

Donald Yeh

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Problem 2

In Exercises 1 through 6 , discuss the continuity of $f$.
$f(x, y)=\left\{\begin{array}{ll}\frac{x^{2} y}{x^{4}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right.$ (HINT: See Example 4, Sec. 19.2.)

Donald Yeh

Numerade Educator

Problem 3

In Exercises 1 through 6 , discuss the continuity of $f$.
$f(x, y)= \begin{cases}\frac{x+y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

Donald Yeh

Numerade Educator

Problem 4

In Exercises 1 through 6 , discuss the continuity of $f$.
$f(x, y)= \begin{cases}\frac{x+y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

Donald Yeh

Numerade Educator

Problem 5

In Exercises 1 through 6 , discuss the continuity of $f$.
$f(x, y)= \begin{cases}\frac{x y}{|x|+|y|} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

Donald Yeh

Numerade Educator

Problem 6

In Exercises 1 through 6 , discuss the continuity of $f$.
$f(x, y)= \begin{cases}\frac{x^{2} y^{2}}{\left|x^{3}\right|+\left|y^{3}\right|} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$

Donald Yeh

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Problem 7

Prove Theorem 19.3.7.

Nick Johnson

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Problem 8

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{y}{\sqrt{x^{2}-y^{2}-4}}

Amy Jiang

Numerade Educator

Problem 9

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x y}{\sqrt{16-x^{2}-y^{2}}}

Dishary Hossain

Dishary Hossain

Numerade Educator

Problem 10

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x^{2}+y^{2}}{\sqrt{9-x^{2}-y^{2}}}

Amy Jiang

Numerade Educator

Problem 11

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x}{\sqrt{4 x^{2}+9 y^{2}-36}}

Dishary Hossain

Dishary Hossain

Numerade Educator

Problem 11

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\frac{x}{\sqrt{4 x^{2}+9 y^{2}-36}}

Dishary Hossain

Dishary Hossain

Numerade Educator

Problem 12

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\ln \left(x^{2}+y^{2}-9\right)-\ln \left(1-x^{2}-y^{2}\right)

Amy Jiang

Numerade Educator

Problem 13

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=x \ln (x y)

Gregory Higby

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Problem 14

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\sin ^{-1}(x y)

Amy Jiang

Numerade Educator

Problem 15

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
f(x, y)=\tan ^{-1} \frac{x}{y}+\sec ^{-1}(x y)

Dishary Hossain

Dishary Hossain

Numerade Educator

Problem 16

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
$$
f(x, y)= \begin{cases}\frac{x^{2}-y^{2}}{x-y} & \text { if } x \neq y \\ x-y & \text { if } x=y\end{cases}
$$

Gregory Higby

Numerade Educator

Problem 17

In Exercises 8 through 17, determine the region of continuity of $f$ and draw a sketch showing as a shaded region in $R^{2}$ the region of continuity of $f$.
$$
f(x, y)= \begin{cases}\frac{\sin (x+y)}{x+y} & \text { if } x+y \neq 0 \\ 1 & \text { if } x+y=0\end{cases}
$$

Amy Jiang

Numerade Educator

Problem 18

In Exercises 18 through 21, prove that the function is discontinuous at the origin. Then determine if the discontinuity is removable or essential. If the discontinuity is removable, define $f(0,0)$ so that the discontinuity is removed.
$$
f(x, y)=\frac{\sqrt{x y}}{x+y}
$$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 19

In Exercises 18 through 21, prove that the function is discontinuous at the origin. Then determine if the discontinuity is removable or essential. If the discontinuity is removable, define $f(0,0)$ so that the discontinuity is removed.
$$
f(x, y)=(x+y) \sin \frac{x}{y}
$$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 20

In Exercises 18 through 21, prove that the function is discontinuous at the origin. Then determine if the discontinuity is removable or essential. If the discontinuity is removable, define $f(0,0)$ so that the discontinuity is removed.
$$
f(x, y)=\frac{x^{2} y^{2}}{x^{2}+y^{2}}
$$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 21

In Exercises 18 through 21, prove that the function is discontinuous at the origin. Then determine if the discontinuity is removable or essential. If the discontinuity is removable, define $f(0,0)$ so that the discontinuity is removed.
$$
f(x, y)=\frac{x^{3} y^{2}}{x^{6}+y^{4}}
$$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 22

(a) Give a definition of continuity at a point for a function of three variables, similar to Definition 19.3.2. (b) State theorems for functions of three variables similar to Theorems $19.3 .3$ and 19.3.7. (c) Define a polynomial function of three variables and a rational function of three variables.

Stanley Enemuo

Numerade Educator

Problem 23

In Exercises 23 through 26, use the definitions and theorems of Exercise 22 to discuss the continuity of the given function.
$$
f(x, y, z)=\frac{x z}{\sqrt{x^{2}+y^{2}+z^{2}-1}}
$$

Israel Hernandez

Israel Hernandez

Numerade Educator

Problem 24

In Exercises 23 through 26, use the definitions and theorems of Exercise 22 to discuss the continuity of the given function.$$
f(x, y, z)=\ln \left(36-4 x^{2}-y^{2}-9 z^{2}\right)
$$

Israel Hernandez

Israel Hernandez

Numerade Educator

Problem 25

In Exercises 23 through 26, use the definitions and theorems of Exercise 22 to discuss the continuity of the given function.
$$
f(x, y, z)= \begin{cases}\frac{3 x y z}{x^{2}+y^{2}+z^{2}} & \text { if }(x, y, z) \neq(0,0,0) \\ 0 & \text { if }(x, y, z)=(0,0,0)\end{cases}
$$

Israel Hernandez

Israel Hernandez

Numerade Educator

Problem 26

In Exercises 23 through 26, use the definitions and theorems of Exercise 22 to discuss the continuity of the given function.
$$
f(x, y, z)= \begin{cases}\frac{x z-y^{2}}{x^{2}+y^{2}+z^{2}} & \text { if }(x, y, z) \neq(0,0,0) \\ 0 & \text { if }(x, y, z)=(0,0,0)\end{cases}
$$

Israel Hernandez

Israel Hernandez

Numerade Educator