Criteria for Pointwise Convergence of Continuous Functions There are many criteria for uniform convergence of continuous functions, such as Stone-Weirestrass etc... However are there any known results guaranteing that a sequence of continuous functions $\{f_n\}$ in $C(R)$ converges pointwise to some discontinuous function?
 A: Generally we can use well known fact, that for convergence, for fixed $x$, $f_n(x)$ should be  Cauchy sequence
For exact condition which gives continuity for sequence continuous functions limit let's consider introduced by C.Arzela so called quasiuniform convergence:
We say, that on some $[a,b]$ segment sequence of continuous functions $f_n(x)$ quasiuniformly converged to continuous function $f(x)$, if for any $\forall \epsilon$ and any $N$ segment $[a,b]$ can be covered by finite amount intervals $(a_1,b_1),(a_1,b_1), \cdots, (a_i,b_i), \cdots, (a_k,b_k)$ and they can be assigned to numbers $n_1,n_2, \cdots, n_i, \cdots, n_k$ $(>N)$ that for every $x$ from $(a_i,b_i)$ performed simultaneously $|f(x)-f_{n_i}(x)|< \epsilon$.
Now using this concept Arzela proved theorem:
Suppose sequence of continuous functions $f_n(x)$ pointwise converged on $[a,b]$ segment to function $f(x)$. Then for $f(x)$ continuity is necessary and sufficient, that $f_n(x)$ converged quasiuniformly.
A: Given $f_n \in C(\mathbb{R})$ converging pointwise to $f$, non-uniform convergence is necessary for   $f \not\in C(\mathbb{R})$ and  uniform  convergence is sufficient for $f \in C(\mathbb{R})$.
However, non-uniform convergence  is not sufficient for the limit function to be discontinuous. For example, with $f_n(x) = n|x| e^{-n|x|}$ we have $\lim_{n \to \infty} f_n(x) = 0$ for all $x$, and the convergence is not uniform since $f_n(1/n) = 1 \not\to 0$ as $n \to \infty$.

Another somewhat related fact is while the pointwise limit of a sequence continuous functions may be discontinuous, the points of discontinuity must form a nowhere dense set.
A: A point-wise limit of a sequence of continuous real-valued functions is commonly called a Baire-1 function.
Some related results:
1.If $X$ is a metric space and $f:X\to \Bbb R$ is a Baire-1 function then the set of points of discontinuity of $f$ is an $F_{\sigma}$ set in X.
2.If $X$ is a metric space with no isolated points and $Y$ is an $F_{\sigma}$ subset of $X$ then there is a Baire-1 function $f:X\to \Bbb R$ with $f^{-1}\{0\}=X\setminus Y$ and such that $Y$ is the set of points of discontinuity of $f$.


*If $f:\Bbb R\to\Bbb R$ is a Baire-1 function then every $f^{-1}\{y\}$ is a $G_{\delta}$ set in $\Bbb R$ and $f^{-1}U$ is  $F_{\sigma \delta}$ in $\Bbb R$ for every open $U\subset \Bbb R$.

