Question

Discuss the continuity of
f(x, y, z)= xy sin z​

Answer 1

The function
`\( f(x, y, z) = xy \sin(z) \)` is continuous for all real numbers
`\( x \), \( y \), and \( z \)`. The continuity of
`\( f(x, y, z) \)` is guaranteed by the continuity of its individual components.

The function
`\( f(x, y, z) = xy \sin(z) \)` is a product of three elementary functions:
`\( xy \)` (a product of two variables) and
`\( \sin(z) \)` (the sine function of one variable).

Each of these individual functions is continuous within its domain. Let's discuss the continuity of
`\( f(x, y, z) \)` based on the continuity of its components:

1. xy:

- The product of two continuous functions is continuous. Both x and y are continuous functions of their respective variables.

- Therefore, the function
`\( xy \)` is continuous.

2. sin(z):

- The sine function
`\( \sin(z) \)` is continuous for all real numbers
`\( z \)`.

- Therefore, the function
`\( \sin(z) \)` is continuous.

3.
`\( f(x, y, z) = xy \sin(z) \)`:

- Since
`\( xy \)` and
`\( \sin(z) \)` are both continuous, their product
`\( xy \sin(z) \)` is also continuous.

- The composition of continuous functions remains continuous.