$\sup\{\frac{1}{f(x)}|x\in[a,b]\} = \frac{1}{\inf\{f(x)|x\in[a,b]\}}$: but is it?
99485
Prove that $1/f$ is Riemann integrable on $[a,b]$
35556
$f(z)=\frac{z}{1+|z|} $ Is 1-1?
28007
Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $
152817
Proving $\sup B=\inf A$
93806
$\|f\|_{L^{p}}=\sup\left\{\int_{{\bf R}^{n}}|f(x)g(x)|~dx:\|g\|_{L^{q}}\le1\right\}$
59872
How to prove $\inf(S)=-\sup(-S)$?
30685
Is $f(a)\!=\!0\!=\!f(b)\Rightarrow (x\!-\!a)(x\!-\!b)\mid f(x)\,$ true if $\,a=b?\ $ [Double Factor Theorem]
12523
Supremum of a Continuous Function is Continuous
117640
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
19561
Why is $\frac{1}{\frac{1}{X}}=X$?
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
137511
Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
78471
Closure of $\{\frac{1}{n}\}:=M$
59063
$f(f(x))=f(x)$ question
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 $f(c) = 0$.
15104
Show that $\big|\frac{a-b}{1-\overline{a}b}\big|=1$, when either $|a|=1$ or $|b|=1$.
106830
are $f(f(x))=f(x)$ and $f(x)=x$ equivalent?