Let $f$ be an injective and continous function. Prove that $f$ is monotone.
91852
A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
132824
Does $f$ monotone and $f\in L_{1}([a,\infty))$ imply $\lim_{t\to\infty} t f(t)=0$?
28392
Construct a monotone function which has countably many discontinuities
143122
Showing that the square root is monotone
74502
Proving the bijection of a continuous function $f$ such that $|f(x)-f(y)| \geq \log(1+|x-y|) \forall x,y$ in $\mathbb{R}.$
81488
If a continuous function$f: (X, \mathscr T_X) \to (Y, \mathscr T_Y)$ is injective (Given $Y$ is Hausdorff), show that X is hausdorff
84301
Showing that some Function on the Unit Disc is injective
20273
Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
72006
Suppose $f(x)$ is linear i.e. $f(x + y) = f(x) +f(y) $ and monotone on $[-\infty, +\infty]$, then $f(x) = ax$, a real.
83461
Continuous iff composition with every linear functional is continuous
150554
proof that $f'$ is continuous at $a$
29154
$f$ is continuous and open implies $f$ injective
152563
Set of discontinuity of monotone function is countable
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
53627
Proving continuity of $f$
54729
Prove that $f$ has a fixed point .