If $f$ is defined on $X$ such that every compact set in $X$ is mapped to a compact set. Is $f$ a continuous map?
34344
Is a function that maps every compact set to a compact set continuous?
97761
Show that a proper continuous map from $X$ to locally compact $Y$ is closed
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
158221
Prob. 4 (a), Sec. 20 in Munkres' Topology, 2nd ed: Are these functions continuous in the product, uniform, and box topologies?
102457
for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
136583
Infimum is a continuous function, compact set
24914
Does every continuous map from $\mathbb{Q}$ to $\mathbb{Q}$ extends continuously as a map from $\mathbb{R}$ to $\mathbb{R}$?
152493
Is an expanding map on a compact metric space continuous?
64487
Continuous function and dense set
31747
Uniqueness of continuous extension from $A$ to $\overline{A}$ for maps into a Hausdorff space
4538
Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous?
118851
$X$ be a Hausdorff topological space , let $f:X \to X$ be a continuous map such that $f\circ f=f$ , then is $f(X)$ is closed in $X$ ?
51247
$f$ be a continuous function maps Cauchy into Cauchy. Is $f$ uniformly continuous?
83311
Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
121957
Is $p\mapsto \|f\|_p$ continuous?
65358
Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.