Lipschitz continuous of an integral expression

Question

Let $f(x,y):\mathbb{R}^2\rightarrow[0,+\infty)$ be a continuous function satisfying that:
(1) For any given $x\in \mathbb{R}$, $f(x,y)$ is a probability density function (pdf) of $y$, i.e. $$\int_{-\infty}^{+\infty}{f(x,y)dy}=1$$ holds for any fixed $x\in \mathbb{R}$.
(2) $f(x,y)$ satisfies Lipschitz condition, i.e. $\exists L>0$, such that $\forall x_1,x_2,y_1,y_2 \in \mathbb{R}$, $$|f(x_1,y_1)-f(x_2,y_2)| \leq L \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$ holds.

My question is, is the following proposition true? If true, how to prove it? If false, what other conditions are required?
Proposition: $\forall a \in \mathbb{R}, \exists L_0>0$, such that $\forall x_1,x_2 \in \mathbb{R}$, $$|\int _a ^{+\infty}f(x_1,y)dy-\int _a ^{+\infty}f(x_2,y)dy| \leq L_0|x_1-x_2|$$ holds.