Answer
Differentiable everywhere
Work Step by Step
We are given that
\[
x^{2} \sin y=f(x, y)
\]
We have to prove that $f(x, y)$ is differentiable everywhere. Thus, obtaining partial derivatives of $f(x, y)$:
\[
\begin{array}{l}
2 x \sin y=f_{x}(x, y) \\
x^{2} \cos y=f_{y}(x, y)
\end{array}
\]
As we can see that $f_{x}$ and $f_{y}$ are both continuous functions $\forall(x, y) \in \mathbb{R},$ by theorem 13.4.4, we can write if both partial derivatives exist and are continuous, then the function is differentiable for all points in the domain.
