 # Master Continuity Equation with Plainmath

Recent questions in Continuity MMDCCC50m 2022-11-19

## Given a vector space $V$, is it possible to endow it with two nonequivalent norms $‖\cdot {‖}_{1}$ and $‖\cdot {‖}_{2}$ such that any linear functional on $V$ is continuous in the sense of $‖\cdot {‖}_{1}$ if only if it is continuous in the sense of $‖\cdot {‖}_{2}$?By nonquivalent norms I mean the induced topologies of the norms are different. drogaid1d8 2022-11-17

## What does continuity mean? perlejatyh8 2022-11-08

## let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f\left(0\right)\le 1$ and for all real $x$, $f\left(x{\right)}^{2}-3f\left(x\right)+2\ge 0$. Prove that $f\left(x\right)\le 1$ for all real $x$.I think you would start by factorising to get $\left(f\left(x\right)-2\right)\left(f\left(x\right)-1\right)\ge 0$ and then $f\left(x\right)\ge 1$ but I'm not really sure where to go from there? I think maybe you can use the intermediate value theorem but I'm not exactly sure how. TIA Aleah Avery 2022-11-03

## The function $f\left(x\right)=\frac{{x}^{2}+x}{x}$ is defined and continuous for all x except x = 0. What value of x must be assigned to f(x) for x = 0 in order that the function be continuous at x = 0? Aldo Ashley 2022-10-30

## How do you prove that the function $x\cdot \frac{x-2}{x-2}$ is not continuous at x=2? gasavasiv 2022-10-30

## So I was studying topology and I came across the next theorem:A function $f:X\to Y$ is continous iff for every $ϵ>0$ there is $\delta >0$ such that if ${d}_{x}\left(x,y\right)<\delta$ then ${d}_{y}\left(f\left(x\right),f\left(y\right)\right)<ϵ$.Since every distance in the uniform topology is at most 1, the theorem will always be true for any function from ${R}^{\omega }$ to ${R}^{\omega }$ with the uniform topology, am I wrong? Chelsea Pruitt 2022-10-27

## What is the definition of continuity at a point? Paola Mayer 2022-10-27

## Let us consider the identity function$f:\left(\mathbb{R},d\right)\to \left(\mathbb{R},{d}_{usual}\right)$$f:x\to x$Here we are considering $d\left(x,y\right)=|\left(x{\right)}^{3}-\left(y{\right)}^{3}|$Is the function $f$ uniformly continuous on closed and bounded interval?I am looking for an example of function $f$ which is not uniformly continuous on a closed and bounded interval but it is continuous. robbbiehu 2022-10-21

## Let the function $f:\mathbb{R}\to \mathbb{R}$ be discontinuous at c. Then the statement: $\mathrm{\forall }ϵ>0,\mathrm{\exists }\delta \in \mathbb{R},\mathrm{\forall }x\in \mathbb{R}\left(|x-c|<\delta \phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}|f\left(x\right)-f\left(c\right)|<ϵ\right)$ is false. The negation of the statement: $\mathrm{\exists }ϵ>0,\mathrm{\forall }\delta \in \mathbb{R},\mathrm{\exists }x\in \mathbb{R}\left(|x-c|<\delta \phantom{\rule{thickmathspace}{0ex}}and\phantom{\rule{thickmathspace}{0ex}}|f\left(x\right)-f\left(c\right)|⩾ϵ\right)$ is false because whenever $\delta$ is negative, $|x-c|<\delta$ is false. Is anything wrong here? Thank you! Emilio Calhoun 2022-10-20

## How do you prove that the function $x\mathrm{sin}\left(\frac{1}{x}\right)$ is continuous at x=0? raapjeqp 2022-10-18

## How do you use the definition of continuity and the properties of limits to show that the function $g\left(x\right)=\sqrt{-{x}^{2}+8\cdot x-15}$ is continuous on the interval [3,5]? Payton George 2022-10-12

## What is the continuity of $f\left(t\right)=3-\sqrt{9-{t}^{2}}$ Antwan Perez 2022-10-12

## How do you prove that the function $\frac{1}{x}$ is continuous at x=1? timberwuf8r 2022-10-07

## What are the three conditions for continuity at a point? Denisse Fitzpatrick 2022-09-30

## I have a rational function $f\left(x\right)=1/\left({x}^{2}-4\right)$. We know that f(x) is not defined at x=2 and x=−2 and has an infinite discontinuity at these x-values. However, I wanted to know if the function is continuous on the interval (0,2] because we know that it is approaching $-\mathrm{\infty }$ as x approaches 2 but if we only have the interval (0,2], it is continuously going to negative infinity. So, is this function continuous in this interval or not? Thank you so much. Krish Crosby 2022-09-26

## For a continuous function (let's say f(x)) at a point x=c, is f(c) the limit of the function as x tends to c? madeeha1d8 2022-09-23

## What makes a function continuous at a point? Jackson Garner 2022-09-18

## Let$f\left(x\right)=\left\{\begin{array}{lcc}\frac{{z}^{3}-1}{{z}^{2}+z+1}& if& |z|\ne 1\\ \\ \frac{-1+i\sqrt{3}}{2}& if& |z|=1\end{array}$is $f$ continous in ${z}_{0}=\frac{1+\sqrt{3}i}{2}$I think to $f$ is not continous at ${z}_{0}$, i try using the sequence criterion for continuity searching a sequence $\left\{{z}_{n}\right\}$ such that $\underset{n\to \mathrm{\infty }}{lim}{z}_{n}={z}_{0}$ but $\underset{n\to \mathrm{\infty }}{lim}f\left({z}_{n}\right)\ne f\left({z}_{0}\right)$. but i cant find that sequence, ill be very grateful for any hint or help to solve my problem. Marcus Bass 2022-09-18
## Consider the space ${\mathbb{R}}^{\mathbb{R}}$ of functions from the set of real numbers to the set of real numbers. A real number $r$ is said to be a period of a given function $f$ iff for all real numbers $x$, $f\left(x+r\right)=f\left(x\right)$. I define a periodic set to be the set of periods of a given function. I define a continuous periodic set to be the set of periods of a given continuous function. It is easy to show that $\mathbb{R}$ itself and also $r\ast \mathbb{Z}$ are continuous periodic sets, where the latter denotes the product of the real number $r$ with the set of integers. Is the converse true? That is, is every continuous periodic set either $\mathbb{R}$ or $r\ast \mathbb{Z}$? Zachariah Norris 2022-09-15