Let $x_{n}$ be a sequence of continuous functions uniformly convergent to the function $x$, the domain of all functions $x_{n}$ be an interval $[a,b]$ and let $g_{n}(t):=f(t,x_{n}(t))$ and $g(t):=f(t,x(t))$, where $f$ is real-valued function with open domain.
Question: What assumptions about $f$ should be made to get the uniform convergence $g_{n}(t)$ to $g(t)$ on the interval $[a,b]$?
I suspect that continuity of $f$ is not sufficient, probably one can assume the uniform continuity of $f$, but I can not prove any of these claims.
I would be very grateful for any hints.