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Chapter 11: Problem 51

In Exercises \(47-52,\) discuss the continuity of the function. \(f(x, y, z)=x y \sin z\)

Short Answer

Expert verified
The function \(f(x, y, z) = x y \sin z\) is continuous everywhere in its domain.

Step by step solution

01

Identify the function

The given multivariable function is \(f(x, y, z)= x y \sin z\).
02

Apply the continuity test

Check the limit of the function as (x, y, z) approaches any arbitrary point (h, k, l), and compare it with the function value at (h, k, l).
03

Calculate the limit

Calculate \(\lim_{{(x, y, z) \rightarrow (h, k, l)}} f(x, y, z) = \lim_{{(x, y, z) \rightarrow (h, k, l)}} x y \sin z = h k \sin l\).
04

Compute the function value

Now, compute \(f(h, k, l) = h k \sin l\).
05

Compare the limit with the function value

Observing the results from the last two steps, we see that \(\lim_{{(x, y, z) \rightarrow (h, k, l)}} f(x, y, z) = h k \sin l = f(h, k, l)\), therefore by the definition of continuity, the function is continuous everywhere in its domain.

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