If I define $f:X\rightarrow Y$ to be continuous on a subset $S\subset X$ if for any sequence $\{x_n\}\in S$ such that $x_n\rightarrow x$ implies that $f(x)\in f(S)$.
My question (and it could be a very dumb one, i just can't see why) is: if for any sequence $\{x_n\}\notin S$ such that $x_n\rightarrow x$ it means that $f(x)\notin f(S)$?
If there's an answer to that already, sorry, but i did not find it.
The answer is negative. Take, for instance $f\colon\mathbb{R}\longrightarrow\mathbb{R}$ defined by $f(x)=x$. Then, bu your definition, $f$ is continuous on $[0,1]$. Now, consider the sequence $\left(1+\frac1n\right)_{n\in\mathbb N}$. Its elements don't belong to $[0,1]$. However, it converges to $1$ and $f(1)\in[0,1]$.