$f:\mathbb{R}\rightarrow \mathbb{R}$ continuous and $f(\mathbb{R}$) is countable, then $f$ is constant.
132478
Prove that continuous functions mapping irrationals to rationals must be constant
86472
$f:\mathbb R \to \mathbb R^n$ be such that $G(f):=\{(x,f(x)):x \in \mathbb R\}$ is closed and connected in $\mathbb R^{n+1}$ , is $f$ continuous ?
7876
If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$
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 $\mathbb{R}$
16413
Proving existence of surjective $f:\mathbb{N} \rightarrow A$ implies $A$ is at most countable.
13765
Is a rational-valued continuous function $f\colon[0,1]\to\mathbb{R}$ constant?
28399
Showing $f$ is constant
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
157821
Let $S$ be a finite set. Show that any function $f: S \rightarrow \mathbb{R}^n$ is continuous on S. What if $S$ is countably infinite?
80643
Show that f(x) defined on [0,infinity) R to is continuous at 0.
152784
$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
124779
Intuition for uniform continuity of a function on $\mathbb{R}$
154034
Suppose that $f(x)$ is continuous on $(0, \infty)$ such that for all $x > 0$,$f(x^2) = f(x)$. Prove that $f$ is a constant function.
2093
If $f$ is continuous then $G$ is homeomorphic to $X$.
44144
If $f:[a,b]\to \mathbb R$ is continuous then so is $\max_{t\in [a,x]}f(t)$?
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.