# Is shifting a continuous function a “(pointwise) continuous process”

$\def\R{\mathbb R}$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.

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Yes, $f$ is continuous hence $\lim_{n \to \infty} f(x + x_n) = f( \lim_{n \to \infty} x + x_n) = f(x)$.