# CRN 11378: HW 5

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'''Problem 21.''' Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$. | '''Problem 21.''' Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$. | ||

# Show that $f$ is continuous at $0$. | # Show that $f$ is continuous at $0$. | ||

− | # Show that $f$ is continuous at any $ | + | # Show that $f$ is continuous at any $x_0\neq 0$. (The identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ will be helpful.) |

− | '''Problem | + | '''Problem 22.''' Assume $f:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$. Show that $\{x\in\mathbb{R}\ |\ f(x)=0\}$ is a closed set. |

− | '''Problem | + | '''Problem 23.''' Let $0<c<1$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$. |

− | + | #Show that $f$ is continuous on $\mathbb{R}$. | |

− | + | #Show that there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y)))\ldots$. | |

− | '''Problem | + | '''Problem 24.''' |

− | + | #Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\mathbb{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$. | |

− | + | #Show that the result fails without the boundedness condition. | |

− | + | ||

− | + |

## Revision as of 23:05, 4 November 2019

**Problem 21.** Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.

- Show that $f$ is continuous at $0$.
- Show that $f$ is continuous at any $x_0\neq 0$. (The identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ will be helpful.)

**Problem 22.** Assume $f:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$. Show that $\{x\in\mathbb{R}\ |\ f(x)=0\}$ is a closed set.

**Problem 23.** Let $0<c<1$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.

- Show that $f$ is continuous on $\mathbb{R}$.
- Show that there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y)))\ldots$.

**Problem 24.**

- Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\mathbb{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$.
- Show that the result fails without the boundedness condition.