Suppose $f$ satisfies
Question
Let $t_{n}$ be a sequence such that $\lim_{n\to\infty} t_{n}=t_{0}$. Does the sequence $f_{n}(x):=f(x,t_{n})$ converge uniformly to $f(x,t_{0})$?
I would like to know if this is the case, because I am trying to understand when I can interchange $\lim$ and $\int$. If the above is true, then I think I could use the Lebesgue convergence theorem to show that $\lim_{x\to x_{0}} \int f=\int\lim_{x\to x_{0}}f$.
What also puzzles me is that I have found a proof of $\lim_{x\to x_{0}} \int f=\int\lim_{x\to x_{0}}f$ that uses only the locally Lipschitz property from above.
Many thanks in advance!