Sequence definition of continuity for function on $\mathbb R^\mathbb N$

Question

The usual definition for continuous function $f:A\to\mathbb R$ is: if $x_n$ converge to $x$ then $f(x_n)$ converge to $f(x)$.

Let $A=\mathbb R^\mathbb N$, what will be the sequence definition of continuity be like?

I am guessing that we have to assume a strong condition that:

if $x_n$ pointwise converge to $x$ then $f(x_n)$ converge to $f(x)$.

For example, the following definition is too weak to make $f$ continuous:

$x_n$ uniformly converge imply $f(x_n)$ converge.

Am I correct? I could be wrong.

The key point of this question is the epsilon-delta definition of continuity stays the same, but we have to extra careful when considering the sequence definition.