If for every sequence $(x_n)$, $x_n \to x_0 \implies (f(x_n))$ is a Cauchy sequence, then is $f$ continuous?
134261
for every convergent sequence $x_n$, $f(x_n)$ also converges. Does this imply continuity of f?
84092
If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?
154521
Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
98762
$f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$?
47381
If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
98599
If a subsequence of a Cauchy sequence converges, then the whole sequence converges.
147841
Please help prove that $(x_n)$ is a Cauchy sequence if $|x_{n+1} - x_n| \leq Cr^n$
110310
$|x_{n+1} - x_n| \le C |x_n - x_{n-1}|.$ Prove $(x_n)$ is cauchy
108057
If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous
46101
If $f(x)$ is continuous on $[a,b]$ and $M=\max \; |f(x)|$, is $M=\lim \limits_{n\to\infty} \left(\int_a^b|f(x)|^n\,\mathrm dx\right)^{1/n}$?
33304
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?
54982
How to prove that $x_n=1+\frac{1}{2!}+\cdots+\frac{1}{n!}$ is a Cauchy sequence?
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
40915
Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
68143
The set of convergent points of a sequence of continuous functions is Borel
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
143618
If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself)