How can I show the integral of two continuous functions results in a Lipschitz continuous function?

Let's say we have continuous function $f: [a,b] \to \mathbb{R}$ and Lipschitz continuous function $g: \mathbb{R} \to \mathbb{R}$. If $h(x) = \int_{a}^{b}g(t-x)f(t)dt$, how can I prove $h$ is Lipschitz continuous?

So our end goal is to show that $|h(b)-h(a)| < C|b-a|$ for some $C$ that is a constant. So I think that perhaps we can start by saying that since $g$ is Lipschitz continuous, it follows that $g(t - x)$ using the Linear property is integrable? I really don't know where I can begin. I do know that the Linear property and the Absolute property will be helpful.

$|h(x)-h(y)|=|\int_{a}^{b} g(t-x)f(t) \, \mathrm{d}t-\int_{a}^{b} g(t-y)f(t) \, \mathrm{d}t|\\ =|\int_{a}^{b} (g(t-x)-g(t-y))f(t) \, \mathrm{d}t |\\\le \int_{a}^{b} |g(t-x)-g(t-y)||f(t)| \, \mathrm{d}t \ \\\le C_1 \int_{a}^{b}|t-x-(t-y)||f(t)| \, \mathrm{d}t \\\le C_1\int_{a}^{b}|x-y||f(t)| \, \mathrm{d}t \\\le C_1|x-y|\int_{a}^{b} |f(t)| \, \mathrm{d}t \\\le C|x-y|.$