Let $(M, \text{d})$ be a metric space, $f:M \rightarrow \mathbb{R}$ and $x_0 \in M$. Suppose that there exists a constant $L$ such that for all $x \in M$
$$ f(x) - f(x_0) \leq L \cdot \text{d}(x, x_0) $$
Does this property have a name? I am thinking of "Local upper Lipschitz semicontinuity" but I can't find something similar.