Answer
Differentiable everywhere
Work Step by Step
We are given that
\[
x y \sin z=f(x, y)
\]
We find:
\[
\begin{array}{l}
y \sin z=f_{x}(x, y) \\
x \sin z=f_{y}(x, y) \\
x y \cos z=f_{y}(x, y)
\end{array}
\]
As we can see that $f_{x}, f_{y}$ and $f_{z}$ are continuous functions $\forall(x, y) \in \mathbb{R},$ by theorem 13.4.4, we can say that if all the partial derivatives of function $f(x, y, z)$ exist and are continuous, then the function is differentiable everywhere.
