 Solution

Fix $t_0\ge 0$. On a compact parameter interval containing all nonnegative $t$ sufficiently close to $t_0$, the corresponding rectangle with $x\in [0,1]$ is compact. The continuous function $f$ is therefore bounded there by some $M<\infty$, by the extreme value theorem.

For any sequence $t_n\ge 0$ tending to $t_0$, all sufficiently late $t_n$ lie in this rectangle's parameter range. For each $x\in [0,1]$, continuity givesThe constant function $M$ is integrable on $[0,1]$. The dominated convergence theorem therefore givesThe sequential criterion for continuity on a real interval proves $F$ is continuous on $[0,\infty )$. At zero this is right-continuity. This is continuity of parameter integrals on compact rectangles; the bound is local in the parameter, so $f$ need not be bounded on the entire unbounded strip.