Continuity of a function on $\mathbb{R}^n\times\mathbb{R}^n$

Question

Given a function $F=F(x,p):\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}$ and provided that

1. for each $p \in \mathbb{R}^n$ $F(\cdot,p)$ is continuous.
2. exist $L>0$ such that $|F(x,p_1)-F(x,p_2)|\leq L|p_1-p_2|$ for all $x,p_1,p_2 \in \mathbb{R}^n$,

can I state that $F$ is continuous on $\mathbb{R}^n\times\mathbb{R}^n$?

Answer 1

Let $\epsilon > 0$, and let $(x_0, p_0)$ be arbitrary. Choose $\delta_1 > 0$ such that $|F(x, p_0) - F(x_0, p_0)| < \epsilon/2$ for $|x - x_0| < \delta_1$. Then $$ $|F(x, p) - F(x_0, p_0)| \le |F(x, p) - F(x, p_0)| + |F(x, p_0) - F(x_0, p_0)|, $$ where $$ |F(x, p) - F(x, p_0)| \le L|p - p_0|. $$ So if $|x - x_0| < \delta_1$ and $|p - p_0| < \delta_2$ where $\delta_2 = \frac{\epsilon}{2L}$ we have $$ $|F(x, p) - F(x_0, p_0)| < \epsilon, $$ proving continuity.