Show that exists a $x\in I$ such that $f(x)=x$
126842
Fixed point in a continuous function
19146
Showing that $f$ has exactly one fixed point
99906
If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
18950
Show that $f$ has at most one fixed point
93771
Prove that for any set $A$ there is some $x \notin A$
60348
How do I prove $F(a)=F(a^2)?$
66951
Proving the existence of a sequence of functions converging to 0 with an added property
11322
Show that $f$ vanishes identically.
64905
Let $f \in C [a,b]$ such that $\forall x\exists y$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$. Prove that $\exists c\in [a,b]$ such that $f(c) = 0$.
1821
Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
82567
Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
80421
Show that $a - b \mid f(a) - f(b)$
101778
Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
148247
Prove the existence of an analytic function