If f:X→R is a continuous non constant function and X is connected then X is uncountable.
39351
$f(X)$ is uncountable and hence $X$ is uncountable.
152563
Set of discontinuity of monotone function is countable
97619
Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
83461
Continuous iff composition with every linear functional is continuous
100740
proof of that a continuous function has a fixed point
34940
Continuity of $f(x) = x^n$
55368
$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly
40209
$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
154521
Linear functional $f$ is continuous at $x_0=0$ if and only if $f$ is continuous $\forall x\in X$?
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 $\mathbb{R}$
117502
Proving that a certain continuous function is surjective.
97316
Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
64487
Continuous function and dense set
145017
Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points
76607
Show that exists a continuous function in $E$ that is not bounded
124444
If $f$ s discontinuous at $x_0$ but $g$ is continuous there, then $f+g$?