Let $S$ be the set of real sequences, and $S_0\subset S$ be the set of convergent real sequences. For each $f:\mathbf R\to\mathbf R$, we define $\bar{f}:S\to S$ by $$\bar{f}(\{x_n\})=\{f(x_n)\}.$$ It is known that

if $f$ is continuous, then for each $\{x_n\}\in S_0$, $\{\lim f(x_n)\}\in S_0$ and $\bar{f}(\lim x_n)=\lim f(x_n).$ ----(#)

So in some sense, $f$ and $\lim$ commute for continuous $f$.

My question is, can we reverse the process, starting with the definition of continuous functions and use this commutativity to define the notion of the limit of a sequence? In other words, is there a unique set $S_0\subset S$ and a unique function $\lim:S_0\to\mathbf R$ such that (#) holds?

Let $S$ be the set of real sequences, and $S_0\subset S$ be the set of convergent real sequences. For each $f:\mathbf R\to\mathbf R$, we define $\bar{f}:S\to S$ by $$\bar{f}(\{x_n\})=\{f(x_n)\}.$$ It is known that

if $f$ is continuous, then for each $\{x_n\}\in S_0$, $\{\lim f(x_n)\}\in S_0$ and $\bar{f}(\lim x_n)=\lim f(x_n).$ ----(#)

So in some sense, $f$ and $\lim$ commute for continuous $f$.

My question is, can we reverse the process, starting with the definition of continuous functions and use this commutativity to define the notion of the limit of a sequence? In other words, is there a largest set $S_0\subset S$ (i.e. contains all other possible $S_0$) and a unique function $\lim:S_0\to\mathbf R$ such that (#) holds?