Let $f: \mathbb R^d \to \mathbb R^d$ be a measurable function such that
$$
\sup_{x,y \in \mathbb R^d} \frac{|f(x) - f(y)|}{\max \{1, |x-y| \}} < \infty.
$$

>Are there functions $g,h: \mathbb R^d \to \mathbb R^d$ such that $f = g + h$ and
$$
\begin{align*}
\sup_{\substack{x,y \in \mathbb R^d \\x \neq y}} \frac{|g(x) - g(y)|}{|x-y|} + \sup_{x \in \mathbb R^d} |h(x)| &< \infty.
\end{align*}
$$
?

Thank you so much for your elaboration!