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

What makes a function continuous at a point?

Jackson Garner 2022-09-18