Problem Set - Continuous functions

## Problem Set - Continuous functions

 Let $f:ℝ\to ℝ$ be such that $f$ is continuous at $x=0$ and if $x,y\in ℝ$ then $f\left(x+y\right)=f\left(x\right)f\left(y\right)$. Show that if $a\in ℝ$ then $f$ is continuous at $x=a$. Let $f:{ℝ}_{>0}\to ℝ$ be such that $f$ is continuous at $x=1$ and if $x,y\in {ℝ}_{>0}$ then $f\left(xy\right)=f\left(x\right)+f\left(y\right)$. Show that if $a\in {ℝ}_{>0}$ then $f$ is continuous at $x=a$. Let $I$ be an interval in $ℝ$. Let $f:I\to ℝ$ be continous. Show that the function $|f|:I\to ℝ$ given by $|f|\left(x\right)=|f\left(x\right)|$ is continuous. Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ and $g:I\to ℝ$ be continous. Show that the function $\mathrm{max}\left(f,g\right):I\to ℝ$ given by $\mathrm{max}\left(f,g\right)\left(x\right)=\mathrm{max}\left(f\left(x\right),g\left(x\right)\right)$ is continuous. (Thomae's function) Let $f:\left[0,1\right]\to ℝ$ be given by $f\left(x\right)=\left\{\begin{array}{cc}\frac{1}{n},& \text{if}\phantom{\rule{0.5em}{0ex}}\frac{m}{n}\in ℚ\phantom{\rule{0.5em}{0ex}}\text{is reduced,}\hfill \\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}x\notin ℚ.\hfill \end{array}$ Show that (a)   If $a\notin ℚ$ then $f$ is continuous at $x=a$, and (b)   If $a\in ℚ$ then $f$ is not continuous at $x=a$. Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ and $g:I\to ℝ$ be continous. Show that the function $\mathrm{min}\left(f,g\right):I\to ℝ$ given by $\mathrm{min}\left(f,g\right)\left(x\right)=\mathrm{min}\left(f\left(x\right),g\left(x\right)\right)$ is continuous. Let $a\in ℝ$ and let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\left\{\begin{array}{cc}ax,& \text{if}\phantom{\rule{0.5em}{0ex}}x\le 0,\hfill \\ \sqrt{x},& \text{if}\phantom{\rule{0.5em}{0ex}}x>0.\hfill \end{array}$ Show that $f$ is continuous. Is the function $f:ℝ\to ℝ$ given by $f\left(x\right)=x$ uniformly continuous? Is the function $f:\left(0,1\right)\to ℝ$ given by $f\left(x\right)=\frac{1}{x}$ uniformly continuous? Is the function $f:\left({10}^{-4},1\right)\to ℝ$ given by $f\left(x\right)=\frac{1}{x}$ uniformly continuous? Is the function $f:\left(0,1\right)\to ℝ$ given by $f\left(x\right)={x}^{2}$ uniformly continuous? Is the function $f:\left[-1,1\right]\to ℝ$ given by $f\left(x\right)=\sqrt{1-{x}^{2}}$ uniformly continuous? Is the function $f:\left(1,\infty \right)\to ℝ$ given by $f\left(x\right)=\mathrm{log}x$ uniformly continuous? Is the function $f:\left(0,\infty \right)\to ℝ$ given by $f\left(x\right)=\mathrm{log}x$ uniformly continuous? Let $f:ℝ\to ℝ$ be the function given by $f\left(x\right)=\frac{x}{\left(1+|x|\right)}$. Show that (a)   $f$ is continuous, (b)   $f$ is uniformly continuous, (c)   $\mathrm{sup}\left(f\left(ℝ\right)\right)=1$, (d)   There does not exist $x\in ℝ$ such that $f\left(x\right)\right)=1$, (e)   $\mathrm{inf}\left(f\left(ℝ\right)\right)=-1$, (d)   There does not exist $y\in ℝ$ such that $f\left(y\right)\right)=-1$. Show that the function $f:ℝ\to ℝ$ given by $f\left(x\right)={x}^{3}-6x+3$ has exactly 3 roots. Let $I$ be an interval in $ℝ$ and let $f:I\to ℝ$ be a continuous function. Prove that $f\left(I\right)$ is an interval. Let $I$ and $J$ be intervals in $ℝ$ and let $f:I\to J$ be a surjective strictly monotonic continuous function. Prove that the inverse function $g:J\to I$ exists and is strictly monotonic and continuous.