If $f:[0,1]\rightarrow [0,1]$ is increasing function. Suppose $0<f(0),f(1)<1$. Show $f$ has fixed point.
35133
Every increasing function from a certain set to itself has at least one fixed point
17234
Show that a continuous function has a fixed point
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
77943
If $f$ is continuous, then there exists $x+\frac{1}{n}\in[0,1]$ and $f\left(x+\frac{1}{n}\right)=f(x)$
120406
$f$ strictly increasing does not imply $f'>0$
108115
If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
48764
Increasing real valued function whose image set is connected
19146
Showing that $f$ has exactly one fixed point
6636
Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?
14975
Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$
30860
Prove that $f(x)=0$ in $[-1,1]$.
79301
Could a constant function be considered increasing?
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
113740
Find a uniformly continuous function such that $a_{n+1}=f(a_n)$
36402
Prove that $f(x) = \cos(x)$ has a unique fixed point.
90892
Is a function increasing if the derivative is positive except at one point of an interval?
108060
Suppose $f:[0,\infty]\rightarrow$ to R and has continuous $f'(x)$ and $f''(x)$ f(x) → 0 as x → ∞. Show that f′(x)→0 as x→∞