Suppose we have continuous $f_{n} : D \rightarrow \mathbb{R}$ converging to $f : D \rightarrow \mathbb{R}$ uniformly on $D$. Then, is $f$ continuous?
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
158344
Uniform continuous maps that converge uniformly to a function f. What can we say about f?
132946
Continuous and bounded imply uniform continuity?
108057
If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous
93461
Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.
52922
let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
136011
Must $f$ be continuous?
13257
How to prove this problem about uniform continuity
83204
Is every $G_\delta$ set the set of continuity points of some function $f$?
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
98772
Continuity of a function in $\mathbb{R}$
62723
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
158221
Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
131060
Is $\sqrt{x}$ uniformly continuous in $\mathbb{R}^+$?
23617
Determine whether $f(x)={\sin x \over x}$ is uniformly continuous in $\mathbb R$
69122
If $f,g$ are uniformly continuous prove $f+g$ is uniformly continuous but $fg$ and $\dfrac{f}{g}$ are not