Prove that $f$ has a fixed point .
17234
Show that a continuous function has a fixed point
149390
Fixed point of a Function (monotonically increasing)
149739
Prove: $f(t)\leq t+1$
61884
If $T^n$ is $q$-contractive, $T$ exactly has one fixed point
69219
Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
121771
Proof $f(x)\equiv 0$
148266
Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
138809
Consequences of a fixed point of a bounded, complex analytic function
11158
Proof that a continuous function with $f(x) = f(x^2)$ is constant.
8799
Mapping homotopic to the identity map has a fixed point
72702
Estimate on the derivative at fixed point
35133
Every increasing function from a certain set to itself has at least one fixed point
28399
Showing $f$ is constant
18950
Show that $f$ has at most one fixed point
36402
Prove that $f(x) = \cos(x)$ has a unique fixed point.
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
151958
How to show $f'(0)$ exist and is equal to $1$?