continuouness of functions

Question

i'm having trouble with the following questions:

1)let $f$ be a continuous function in $\mathbb R$. prove that if $|f|$ is monotonous and rising in $\mathbb R$, then $f$ is monotonous in $\mathbb R$. what i tried to do is to first use the characteristics of a continuous function, and by knowing it's well defined in $\mathbb R$ and that $\lim $ where $x\to x_0$ equals $\lim f(x_0)$, i tried to show $|f|$ is raising in $\mathbb R$, thus $f$ is monotonous. but my problem is that i can't actually show how i can conclude that f is monotonous. if you can please show me how to prove it correctly, because i am not sure of the right way or how to prove such things correctly(how should it be written).

2)prove that $ f(x)= \frac{1}{\sin x}+\frac{1}{x-1} $ can obtain in $(0,1)$ every value in $\mathbb R$. here i first tried to find where it is not defined, then tried to find its limits at both $x\to 0,1$ and then tried to show that $\lim_(x\to x_0) f(x) = f(x_0)$. but i think it's totally wrong. please show me how to do it correctly, and show me the right way to write mathmatical proves, so that i can learn the right way to prove claims.

thank you very much for your help.

Answer 1

Hint: (I suppose that your hypothesis is that $|f|$ is strictly increasing) . Show that $f$ is injective and see here A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.

Answer 2

1. Claim. It is impossible that $f$ assumes values of both nonzero signs.

Proof. Assume there are $a<b$ with $f(a)f(b)<0$. By the IVT there is a $c\in\ ]a,b[\ $ with $f(c)=0$. We then would have $|f(a)|>|f(c)|$, contradicting the assumption about $|f|$.$\quad\square$

This allows to conclude that either $f=|f|$ or $f=-|f|$.

2. One has $\lim_{x\to0+} f(x)=\infty$ and $\lim_{x\to1-} f(x)=-\infty$, and $f$ is continuous on $\ ]0,1[\ $. Let a $c\in{\mathbb R}$ be given. We have to prove that there is a $\xi\in\ ]0,1[\ $ with $f(\xi)=c$. To this end establish an $a\in\ ]0,1[\ $ with $f(a)>c$ and a $b\in\ ]0,1[\ $ with $f(b)<c$.