Let $f:[a,b]\mapsto[a,b]$ be continous, suppose $f(a)=f(b)$. Show that there exist $c \in [a,\frac{b+a}{2}]$ with $f(c)=f(c+\frac{b-a}{2})$
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
74364
Show that function $f$ has a continuous extension to $[a,b]$ iff $f$ is uniformly continuous on $(a,b)$
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
152841
I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive.
15262
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
55605
If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
94781
Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
144614
If $f$ is continuous on $[a,b]$ and $F(x) = \sup f([a,x])$. Prove that $F$ is continuous on $[a,b]$ .
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
105769
Prove that $\exists a, b\in(0,1)$ such that $\int_0^{a} xf(x)dx=0\text{ and }\int_0^bxf(x)dx=\frac{b^2f(b)}{2}.$
19457
Show that: $\inf(A+B) = \inf(A)+ \inf(B)$
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
30583
If $a\mid b$ and $b\mid a$, $a,b\in R$, show that $\exists w \in R^\times$ with $b=wa$
30694
Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
52922
let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and..
41201
How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$.