Let $(f_n)$ be a sequence of functions $\mathbb{R} \rightarrow \mathbb{R}$.
Suppose that for any $(x_n)$ convergent to $x$ we have $f_n(x_n) \rightarrow f(x)$.
Prove that $f$ in continuous, there is no assumption about $f_n$'s continuity.
Could you help me with that?
