Suppose $f(0) = f(2 \pi)$. Show there is a number $c \in [0, \pi]$ such that $f(c)=f(c + \pi)$
48476
Mean value theorem rolle's theorem
15262
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
32525
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
106748
Number of functions $f : A \to A$ such that $f(f(x))=f(x)$
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
84313
$|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
45602
Show that is $\int_A f$ exists, then so does $\int_A |f|$
145424
If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
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 $f(x) = f(x+\frac{1}{n})$.
135511
$M_a =\{ f\in C[0,1] |\ f(a)=0 \}$ for $a$ $\in$ $[0,1]$. Is $M_a$ finitely generated in $C[0,1]$?
112512
How to prove that there are finitely many $n$ such that $f(1+1/n) = \frac{1}{n+1}$?
136056
$ f:\mathbb C\rightarrow \mathbb C$ defined by $f(z)=0$, if $Re(z)=0$ or $Im(z)=0$ and $f(z)=z$, otherwise.
146152
Characterising functions $f$ that can be written as $f = g \circ g$?
84510
If $f(\mathbb{C})\subset \mathbb{C}-[0,1]$ then $f$ is constant
138026
To show that $n!f(x) \in \mathbb{Z}[x]$