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 measurable 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!

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!