Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{R}$ satisfying: $$f(M)=g(\sigma_1(M),...,\sigma_n(M))$$ where $M\in\mathbb{R}^{n\times n}$ and $\sigma_i(M)\geq 0$ are singular values of $M$ and $f(0)=g(0)=0$.

Now assume $g$ is Lipschitz in every coordinate, i.e. there exists constant $L>0$ such that for any $i=1,...,n$: $$|g(x_1,...,x_i,...,x_n)-g(x_1,...,x_i^\prime,...,x_n)|\leq L|x_i-x_i^\prime|$$

My question is that does $f$ have some sort of Lipschitz property? For example, for any $M,M^\prime\in\mathbb{R}^{n\times n}$, does there exist a constant $C>0$, such that $$|f(M)-f(M^\prime)|\leq C\|M-M^\prime\|_F$$

On the other direction, does the Lipschitz of $f$ imply the Lipschitz of $g$?