$\def\R{\mathbb R}$If $f\colon \R\to\R$ is a continuous function and $(x_n)$ is a sequence such that $\lim_{n\to\infty}x_n = 0$, then is it true that the sequence of functions $y_n = f(\cdot+x_n)$ has pointwise limit $f$? Obviously $y_n$ is not a uniform convergent sequence of functions, e.g. $f(x) = x^2$ and take any $(x_n)$ with $x_n\to 0$.
I suspect it is true. This one has been bugging me all morning.