Let $f:X \to Y $ a continuous function, where $(X, \tau_X)$ is $T_2$ and first countable and let $D$ dense in X, then $f(D) \cap f(X-D)= \emptyset$
140052
Show that $f(X - D) \cap f(D) = \varnothing$ with $f$ continuous in $X$, $D$ dense in $X$ and $f|_{D}$ homeomorphism
13852
Assume $f$ is a continuous one-to-one function over an interval. Prove that $f$ is strictly monotone
125682
Proof verification: the product of two continuous functions is continuous
51156
Function is Continuous When Restricted to Closed Sets Which Cover the Space
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
41690
Products of Continuous functions: Topology
41477
Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
34344
Is a function that maps every compact set to a compact set continuous?
2142
Finding a set of continuous functions with a certain property 2
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
54600
Is a convex and lower semicontinuous function defined on a closed and convex subset of $\mathbb{R}^n$ continuous?
149095
$f(\mathbb{R}\setminus \mathbb{Q}) \subseteq \mathbb{Q}$ and $f(\mathbb{Q}) \subseteq \mathbb{R}\setminus \mathbb{Q}$ imply that $f$ is not continuous
122190
Kernel of $T$ is closed iff $T$ is continuous
118332
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
23728
Does a continuous and 1-1 function map Borel sets to Borel sets?
42794
Prove that functions map countable sets to countable sets