#### Answer

The domain of $h(x) = xe^{sin~x}$ is $(-\infty, \infty)$, and $h(x)$ is continuous on this domain.

#### Work Step by Step

$sin~x$ is continuous for all real numbers and $e^x$ is continuous for all real numbers.
Then $e^{sin~x}$ is continuous for all real numbers.
The function $f(x) = x$ is continuous for all real numbers.
Therefore the product of $x$ and $e^{sin~x}$ is continuous for all real numbers.
The domain of $h(x) = xe^{sin~x}$ is $(-\infty, \infty)$, and $h(x)$ is continuous on this domain.