When is a general time-varying cost function $U(x, t)$ considered to be convex?

Question

I found the following definition in a thesis but with no reference: A cost function $U(x, t)$ is said to be convex if it satisfies the following inequality: $ U(\alpha x_1 + (1 - \alpha) x_2, t) \leq \alpha U(x_1, t) + (1 - \alpha) U(x_2, t) $ for all $(x_1, x_2 \in \mathbb{R}^n)$, $(0 \leq \alpha \leq 1)$, and $(t \in T)$, where $(T)$ is the time domain. In other words, for a general time-varying function $U(x, t)$, $U(x, t)$ is said to be convex if $U(x, t)$ is a convex function of $(x)$ for fixed $(t)$. Is it a valid definition? Can someone provide me with a valid reference where the convexity of a time-varying function is defined and discussed?