Prove that if f '(x) <= 1/2 for all x in the domain of f, then f has at most one fixed point.
39321
If $f:\mathbb R \to \mathbb R$ satisfies $f'(x) <L$ for some $L<1$, then $f$ has a unique fixed point
36402
Prove that $f(x) = \cos(x)$ has a unique fixed point.
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
81758
Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero.
85739
Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
62134
Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots
126842
Fixed point in a continuous function
101570
If derivative of function is less than or equal to 4, prove the existence of point $c$ so $f(c)=c^2$.
30455
Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$
118048
Can a function with just one point in its domain be continuous?
99827
For what values will f(x) be necessarily one-one?
30860
Prove that $f(x)=0$ in $[-1,1]$.
157960
How can I prove the existence of two values in a given domain Mean value theorem?
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
63526
Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
148266
Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
75653
Prove that $|f ''|\ge 4$