$f$ is continuous in $\mathbb{R}$ if and only if for every open set $G$, the set $f^{-1}(G)=\{x:f(x)\in G\} $ is open.
62118
Prove $\epsilon$-$\delta$ definition of continuity implies the open set definition for real function
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 $X\times Y$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
141223
$f(x^2) = 2f(x)$ and $f(x)$ continuous
83326
$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
144013
If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
92089
If $f:[a,b]\to \mathbb{R}$ is continuous and nonnegative and $\int_a^b{f}=0$, then $f(x)=0$ for all $x\in [a,b]$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
122086
continuous function: preimage of closure of a set is closure of preimage
40915
Proof if $f$ continuous in $x_0$ then there is a neighbourhood of $x_0$ so f bounded.
103932
How can I prove that a polynomial with degree $n$ is continuous everywhere in $\mathbb{R}$ using definitions?
108734
$T$ is continuous if and only if $\ker T$ is closed
124779
Intuition for uniform continuity of a function on $\mathbb{R}$
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.
84512
The preimage of continuous function on a closed set is closed.
143174
property of continuous function