Let $D\subset\mathbb R$ and let $f_n:D\to\mathbb R$ be continuous functions on $D$ and let the sequence $(f_n)_{n\to\mathbb N}$ be uniformly convergent to the limit function $f:D\to\mathbb R.$ Then $f$ is also continuous on $D.$
The limit function $f$ does not have to be continuous if the continous functions $f_n$ converge only pointwise to $f.$
Proofs: 1