If $f:\mathbb R\to \mathbb N$ is continuous, then it's constant.
727
Prove that any continuous integer-valued function of a real variable is constant.
48867
why all constant functions are continuous.$?$
145368
How to show $f:Y\to X^{+}$ is continuous?
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
105652
Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
106060
If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
23436
$f$ Holder continuous with Holder exponent $p>1\implies f \text{ is constant}$
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
24017
Let $f: \Bbb R \to [0,\infty)$ be a continuous function such that $g(x)=(f(x))^2$ is uniformly continuous. Which of the following is always true?
93461
Suppose that $f:\Bbb R\to \Bbb R$ is continuous and that $f(x)\to 0$ as $x\to \pm\infty.$Prove that $f$ is uniformly continuous.
60700
Continuity of a function to the integers
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.
52922
let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
47419
Asymptotics of uniformly continuous functions over $\mathbb R$