How to prove that if $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $|f(x) - f(y)| > |x-y|$ for all $x,y$,then range of $f$ is $\mathbb{R}$
117502
Proving that a certain continuous function is surjective.
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
23617
Determine whether $f(x)={\sin x \over x}$ is uniformly continuous in $\mathbb R$
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
46287
Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
58597
A characterization of functions from $\mathbb R^n$ to $\mathbb R^m$ which are continuous
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
62723
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere
46629
Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
122160
$f:\mathbb R \to \mathbb R$ be a continuous such that $f(f(f(x)))=x,\forall x \in \mathbb R$ , is $f(x)=x , \forall x \in \mathbb R$?
143174
property of continuous 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 $X\times Y$
155618
Slightly changing the formal definition of continuity of $f: \mathbb{R} \to \mathbb{R}$?
155923
continuity of $f(x,y)$ in $\mathbb R^2$.
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