Prove the minimum $m(t,x)$ of $F(t,x,y)$ is a continuous function

$$F(t,x,y)=\frac{(x-y)^2}{2t}+\int_{0}^{y} u_0(\eta) d\eta,$$

$$m(t,x)=\min_{y\in\Bbb R} F(t,x,y),$$

where $(t,x,y)\in\Bbb R^3$, $u_0$ is a measurable function, and $|u_0(\eta)|\le M$.

There is a lemma:If $F(t,x,y)=m(t,x)$, then $x-Mt \leq y \leq x+Mt$.

Question: How to use this lemma to prove the continuity of $m(t,x)$?