Suppose $f_n\rightarrow f$ pointwise. Show that if each $f_n$ is continuous then $f$ is also continuous.
140445
Sequence of continuous functions whose pointwise limit is discontinuous
145919
If $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous at $0$ and $f(x)=f(2x)$ for each $x\in\Bbb{R}$ then $f$ is constant.
44057
Pointwise but not uniform convergence of continuous functions on $[0,1]$
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
152784
$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?
90619
For closed subsets $A,B \subseteq X$ with $X = A \cup B$ show that $f \colon X \to Y$ is continuous iff $f|_A$ and $f|_B$ are continuous.
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
143618
If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
41690
Products of Continuous functions: Topology
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
94737
How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$