$f$ is continuous $\iff$ $\forall S \subseteq X, f(\text{cl}(S)) \subseteq \text{cl}(f(S))$
102457
for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.
98772
Continuity of a function in $\mathbb{R}$
151427
f, g continuous for all rationals follow by continuous on all reals?
118207
Prove that the product of two continuous functions is continuous
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
57642
Uniform Continuity implies Continuity
22061
$|f(x)-f(y)|\geq k|x-y|$.Then $f$ is bijective and its inverse is continuous.
112409
If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
128069
Continuity in Slices + Mapping Compact Sets to Compact Sets = Continuous
157821
Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
108734
$T$ is continuous if and only if $\ker T$ is closed
129073
If $f'$ is bounded, then $f$ is uniformly continuous
9320
Extension of continuous function on a closed subset of $\mathbb{R}$
47199
Can $\le$ be used insted of < in the definition of continuity?
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
94737
How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$