continuity on compact subsets implies continuity
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
76351
Equivalence of continuity definitions
136011
Must $f$ be continuous?
156996
Do there exist general conditions under which we can conclude that continuity on a topological space is detected by $\mathbb{R}$?
29303
Cartesian product of compact sets is compact
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.
155618
Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
34940
Continuity of $f(x) = x^n$
62334
Continuity and Joint Continuity
83831
Continuity based on restricted continuity of two subsets
59519
What about the continuity of these functions in the uniform topology?
106534
"Closed subsets of compact space are compact" with an alternative definition of covering
28637
Proving continuous image of compact sets are compact
44647
Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .