Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and 1-to-1. Show that for each borel $E$, $f(E)$ is borel
23728
Does a continuous and 1-1 function map Borel sets to Borel sets?
36227
Show continuity or uniform continuity of $\phi: (C([0,1];\Bbb R ),||\cdot||_\infty )\to (\Bbb R, |\cdot | )$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
46514
Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic
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 $\forall\alpha\in\mathbb{R}$
14578
Proving that an additive function $f$ is continuous if it is continuous at a single point
110458
Show that if $f$ is continuous on $(0,1)$ and $\lim_{x \to 0}f(x) = + \infty$ then $f$ is not uniformly continuous
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
70645
Borel set preserved by continuous map
97256
For $f$ continuous, show $\lim_{n\to\infty} n\int_0^1 f(x)x^n\,dx = f(1).$
31710
Show that $\mathcal{J}=\{[a, a+1) \mid a \in \mathbb{R}\}$ generates the Borel set of $\mathbb{R}$
108057
If $(f_n)\to f$ uniformly and $f_n$ is uniformly continuous for all $n$ then $f$ is uniformly continuous
125682
Proof verification: the product of two continuous functions is continuous
113687
$f$ is continuous function from $ [0,1]$ to itself and f(0)=0, f(1)=1 and $f(f(x))=x$ then show that $f(x)=x$ on$[0,1]$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
9320
Extension of continuous function on a closed subset of $\mathbb{R}$