Recent questions in Continuity

Differential CalculusAnswered question

MMDCCC50m 2022-11-19

Given a vector space $V$, is it possible to endow it with two nonequivalent norms $\Vert \cdot {\Vert}_{1}$ and $\Vert \cdot {\Vert}_{2}$ such that any linear functional on $V$ is continuous in the sense of $\Vert \cdot {\Vert}_{1}$ if only if it is continuous in the sense of $\Vert \cdot {\Vert}_{2}$?

By nonquivalent norms I mean the induced topologies of the norms are different.

By nonquivalent norms I mean the induced topologies of the norms are different.

Differential CalculusAnswered question

perlejatyh8 2022-11-08

let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function such that $f(0)\le 1$ and for all real $x$, $f(x{)}^{2}-3f(x)+2\ge 0$. Prove that $f(x)\le 1$ for all real $x$.

I think you would start by factorising to get $(f(x)-2)(f(x)-1)\ge 0$ and then $f(x)\ge 1$ 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

I think you would start by factorising to get $(f(x)-2)(f(x)-1)\ge 0$ and then $f(x)\ge 1$ 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

Differential CalculusAnswered question

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?

Differential CalculusAnswered question

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?

Differential CalculusAnswered question

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 $\u03f5>0$ there is $\delta >0$ such that if ${d}_{x}(x,y)<\delta $ then ${d}_{y}(f(x),f(y))<\u03f5$.

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?

A function $f:X\to Y$ is continous iff for every $\u03f5>0$ there is $\delta >0$ such that if ${d}_{x}(x,y)<\delta $ then ${d}_{y}(f(x),f(y))<\u03f5$.

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?

Differential CalculusAnswered question

Chelsea Pruitt 2022-10-27

What is the definition of continuity at a point?

Differential CalculusAnswered question

Paola Mayer 2022-10-27

Let us consider the identity function

$f:(\mathbb{R},d)\to (\mathbb{R},{d}_{usual})$

$f:x\to x$

Here we are considering $d(x,y)=|(x{)}^{3}-(y{)}^{3}|$

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.

$f:(\mathbb{R},d)\to (\mathbb{R},{d}_{usual})$

$f:x\to x$

Here we are considering $d(x,y)=|(x{)}^{3}-(y{)}^{3}|$

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.

Differential CalculusAnswered question

robbbiehu 2022-10-21

Let the function $f:\mathbb{R}\to \mathbb{R}$ be discontinuous at c. Then the statement: $\mathrm{\forall}\u03f5>0,\mathrm{\exists}\delta \in \mathbb{R},\mathrm{\forall}x\in \mathbb{R}(|x-c|<\delta \phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}|f(x)-f(c)|<\u03f5)$ is false. The negation of the statement: $\mathrm{\exists}\u03f5>0,\mathrm{\forall}\delta \in \mathbb{R},\mathrm{\exists}x\in \mathbb{R}(|x-c|<\delta \phantom{\rule{thickmathspace}{0ex}}and\phantom{\rule{thickmathspace}{0ex}}|f(x)-f(c)|\u2a7e\u03f5)$ is false because whenever $\delta $ is negative, $|x-c|<\delta $ is false. Is anything wrong here? Thank you!

Differential CalculusAnswered question

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?

Differential CalculusAnswered question

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]?

Differential CalculusAnswered question

Payton George 2022-10-12

What is the continuity of $f\left(t\right)=3-\sqrt{9-{t}^{2}}$

Differential CalculusAnswered question

Antwan Perez 2022-10-12

How do you prove that the function $\frac{1}{x}$ is continuous at x=1?

Differential CalculusAnswered question

timberwuf8r 2022-10-07

What are the three conditions for continuity at a point?

Differential CalculusAnswered question

Denisse Fitzpatrick 2022-09-30

I have a rational function $f(x)=1/({x}^{2}-4)$. 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.

Differential CalculusAnswered question

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?

Differential CalculusAnswered question

madeeha1d8 2022-09-23

What makes a function continuous at a point?

Differential CalculusAnswered question

Jackson Garner 2022-09-18

Let

$f(x)=\{\begin{array}{lcc}\frac{{z}^{3}-1}{{z}^{2}+z+1}& if& |z|\ne 1\\ \\ \frac{-1+i\sqrt{3}}{2}& if& |z|=1\end{array}$

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 $\{{z}_{n}\}$ such that $\underset{n\to \mathrm{\infty}}{lim}{z}_{n}={z}_{0}$ but $\underset{n\to \mathrm{\infty}}{lim}f({z}_{n})\ne f({z}_{0})$. but i cant find that sequence, ill be very grateful for any hint or help to solve my problem.

$f(x)=\{\begin{array}{lcc}\frac{{z}^{3}-1}{{z}^{2}+z+1}& if& |z|\ne 1\\ \\ \frac{-1+i\sqrt{3}}{2}& if& |z|=1\end{array}$

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 $\{{z}_{n}\}$ such that $\underset{n\to \mathrm{\infty}}{lim}{z}_{n}={z}_{0}$ but $\underset{n\to \mathrm{\infty}}{lim}f({z}_{n})\ne f({z}_{0})$. but i cant find that sequence, ill be very grateful for any hint or help to solve my problem.

Differential CalculusAnswered question

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 $x$, $f(x+r)=f(x)$. 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 $r\ast \mathbb{Z}$?

Differential CalculusAnswered question

Zachariah Norris 2022-09-15

How do you prove that the function $f\left(x\right)={x}^{2}-3x+5$ is continuous at a =2?

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