Suppose $f(x_{1},x_{2},\cdots,x_{n})$ satisfies that $\frac{\partial f}{\partial x_{i}}$ is bounded variation in $\Omega$ for $1\leq i\leq n$. Then $$ f\in L^{\frac{n}{n-1}}\left(\Omega\right). $$
Is this true? If yes, where can I find the statement? Please leave the link or the name of the book. Thanks in advance.
If the domain is bounded and convex then $f\in W^{1,\infty}$, by your assumption (since the derivatives are going to be bounded almost everywhere). So this is true in this case. If the domain is unbounded, there are counter examples like the identity function of $\mathbb{R}^n$.
May be you could check the Gagliardo Sobolev Nirenberg inequality, for example in the book by Evans and Gariepy (I am not sure why you chose the exponent).