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What must be true of a function $f:\mathbb{R}\to \mathbb{R}$ and a sequence $(x_n)_{n \in \mathbb{N}} \to L$ such that $$\lim_{n \to\infty} f(x_n) = f(\lim_{n\to\infty} x_n)$$

I don't know if it's true for any general class of functions but I think it's true for functions continuous is some neighborhood of $L$.

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    It is in general true, as long as the limits you are creating exist. – 2017-02-03
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    It certainly does not hold in general. Take $f(x)$ to be zero everywhere except $f(0) =1$ and let $x_n = 1/n$. Then $x_n \to 0$ but $f(x_n) \not\to f(0)$. – 2017-02-03
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    Or $f$ might not be defined at $\lim x_{n}$. – 2017-02-03

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This is certainly true if $f$ is continuous at $L$. In fact, if $f$ is continuous at $L$, then for any sequence $x_n$ converging to $L$, we see $$\lim_{n\to\infty} f(x_n) = f(L).$$ Indeed, conversely, this is one way to define continuity. The function $f$ is continuous at $x$ if and only if for any sequence $x_n$ such that $x_n \neq x$ for all $n$ and $x_n \to x$, we have $$\lim_{n \to \infty} f(x_n) = f(x).$$