Lipschitz condition for a function indexed by a parameter

Question

Consider a random variable $X:\Omega \rightarrow \mathcal{X}$ with probability distribution $P$. Consider a function of $X$ indexed by a parameter $\theta$, $f(X, \theta)$, with $\theta \in \Theta \subseteq \mathbb{\mathbb{R}^l}$ and $f:\Theta \times \mathcal{X}\rightarrow \mathbb{R}$. Assume $\exists$ $m:\mathcal{X}\rightarrow \mathbb{R}$ such that $$ |f(x, \theta_1)-f(x, \theta_2)|\leq m(x) ||\theta_1-\theta_2|| $$ $\forall \theta_1, \theta_2 \in \Theta$, $\forall x \in \mathcal{X}$. Does this mean that $f(\cdot)$ is Lipschitz in $\theta$? Is this Lipschitz condition "pointwise" in the sense that $m(x)$ varies with $x$?