Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous function and $x_0 \in \mathbb R$ such that f is differentiable on both intervals $(-\infty, x_0]$ and $[x_0, +\infty)$. Prove or disprove that there exist two functions $g, h : \mathbb R \rightarrow \mathbb R$ differentiable everywhere such that
$$ f(x) = g(x) + h(x)|x - x_0|\ \ \forall x \in \mathbb R. $$
This feels like it characterizes every non-differentiable point of a continuous function in terms of absolute values but I couldn't come up with a function to disprove nor I was able to construct $g$ and $h$.
Help and directions appreciated.