If $f:S \to T$ . $f$ is a one-to-one continuous function . $V $ is an open subset of $S$ , can we prove that $f(V)$ is an open subset of $T$?
52798
Finding counterexamples: bijective continuous functions that are not homeomorphisms
121993
Prove that $f$ is one-to-one iff $f \circ h = f \circ k$ implies $h = k$.
85612
Let functions $ f: A \to B$ and $g: B \to A$, suppose that $ g \circ f = i_A$. Prove that if $f$ is onto, then $g$ is one-to-one
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.
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
83326
$f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
30965
$f$ is continuous on $X$ iff $f$ is continuous on every compact subset
53738
Monotone functions and continuity
135454
An open interval is an open set?
143174
property of continuous function
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
15487
Continuous function from a connected set?
44402
If $\{x:f(x) > a\}$ and $\{x:f(x) < a\}$ are open$\forall~a\in\mathbb{R}$ then f is continuous.
41477
Prove $f(S \cap T) \subseteq f(S) \cap f(T)$
139744
For a continuous function $f$ show $\exists c\in (0,1)$ s.t $f(c)=3c^2$
23728
Does a continuous and 1-1 function map Borel sets to Borel sets?
73130
A simple question about open set