Partial derivatives and differentiability, continuity

Question

Function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$ has in every $x$ of domain partial derivatives $\frac{\partial f}{\partial x_1}(x) =x_2$, $\frac{\partial f}{\partial x_2}(x) =x_1$, $\frac{\partial f}{\partial x_3}(x) =x_3$. Determine whether

• a) $f$ is continous
• b) $f$ is differentiable

How to do that without knowing $f$'s formula?

Answer 1

Theorem: $f:\mathbb R^n\to\mathbb R$ is $\mathcal C^1$ (continuously differentiable) $\iff$ $f$ has continuous partial derivatives.

Proof of Sufficiency: Proof that continuous partial derivatives implies differentiability

Conclusion

$f$ is differentiable.