Prove $f$ is continuous if $f(\overline{A}) \subset \overline{f(A)}$ for all subsets $A$.
12490
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
72847
Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere?
62887
Proof that a linear transformation is continuous
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
31741
Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous
145368
How to show $f:Y\to X^{+}$ is continuous?
110112
Proving a function is Lipschitz continuous
152784
$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
55238
Show that if $f:X\to Y$ is a continuous function if and only if the graph of $f$ is a closed subset of $X\times Y$
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
44574
Continuity of absolute value of a function
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
66691
Real function continuous on closed interval implies it is bounded - over-simple proof??
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$