Providing that $f :\Bbb R\to\Bbb R$ is continuous . prove that Range of $f$ is $\Bbb R$ if $|f(x)-f(y)|\ge|x-y|$ .
125644
How to check f is surjective
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
130299
Prove that if a function $f: X\to Y$ continuous then its graph is closed
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 $X\times Y$
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.
47381
If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
37966
If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
93021
$f:X\longrightarrow\mathbb{R}$ is continuous iff $\{x\in X:f(x)\geq\alpha\}$ and $\{x\in X:f(x)\leq\alpha\}$ are closed $\forall\alpha\in\mathbb{R}$
33330
Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$
151427
f, g continuous for all rationals follow by continuous on all reals?
43189
Examples of continuous functions $f : \mathbb{R} \to \mathbb{R}$
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
131992
A problem on continuous functions
9406
Compactness of $(M, d)$ in terms of continuous functions $f:M\to \mathbb R$
108057
If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous