If $f_m$ is a sequence of continuous functions which converges uniformly to a function. then that function is continuous as well.
40297
Proof of uniform limit of Continuous Functions
138815
For which conditions on countable sets does continuity implies uniform continuity
29180
If $f(x/n)\to0$ when $n\to\infty$, for every $x$ and $f$ is continuous, then $f(x)\to0$ when $x\to0$, or not?
140445
Sequence of continuous functions whose pointwise limit is discontinuous
12408
Prove that a function whose derivative is bounded is uniformly continuous.
143618
If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)
154751
$f$ brings convergent nets to convergent nets, is it continuous?
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
108057
If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous
144957
Uniform convergence of a function
101906
Why is this sequence of equicontinuous functions uniformly bounded?
84785
Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$
142645
$f_n$ converges uniformly on $[a,b]$ to some function $f$
15579
Show that $f$ is uniformly continuous if limit exists
100740
proof of that a continuous function has a fixed point
143593
A sequence of differentiable functions that converge uniformly to a non differentiable function - is my example correct?
88610
The set of convergence of a sequence of measurable functions is measurable