Dive into Continuity Equation with Help from Our Experts

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 $f : R \to R$ be a continuous function such that $f (0) \leq 1$ and for all real $x$ , $f (x)^{2} - 3 f (x) + 2 \geq 0$ . Prove that $f (x) \leq 1$ for all real $x$ .
I think you would start by factorising to get $(f (x) - 2) (f (x) - 1) \geq 0$ and then $f (x) \geq 1$ 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 $f (x) = \frac{x^{2} + x}{x}$ 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 $δ > 0$ such that if $d_{x} (x, y) < δ$ then $d_{y} (f (x), f (y)) < ϵ$ .
Since every distance in the uniform topology is at most 1, the theorem will always be true for any function from $R^{ω}$ to $R^{ω}$ 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
$f : (R, d) \to (R, d_{u s u a l})$
$f : x \to x$
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.

robbbiehu 2022-10-21

Let the function $f : R \to R$ be discontinuous at c. Then the statement: $\forall ϵ > 0, \exists δ \in R, \forall x \in R (| x - c | < δ ⟹ | f (x) - f (c) | < ϵ)$ is false. The negation of the statement: $\exists ϵ > 0, \forall δ \in R, \exists x \in R (| x - c | < δ a n d | f (x) - f (c) | ⩾ ϵ)$ is false because whenever $δ$ is negative, $| x - c | < δ$ is false. Is anything wrong here? Thank you!

Emilio Calhoun 2022-10-20

How do you prove that the function $x \sin (\frac{1}{x})$ 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 $g (x) = \sqrt{- x^{2} + 8 \cdot x - 15}$ is continuous on the interval [3,5]?

Payton George 2022-10-12

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

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 $f (x) = 1 / (x^{2} - 4)$ . 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 $- \infty$ 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
$f (x) = {\begin{array}{lcc} \frac{z^{3} - 1}{z^{2} + z + 1} & i f & | z | \neq 1 \\ \frac{- 1 + i \sqrt{3}}{2} & i f & | z | = 1 \end{array}$
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 $lim_{n \to \infty} z_{n} = z_{0}$ but $lim_{n \to \infty} f (z_{n}) \neq f (z_{0})$ . 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 $R^{R}$ 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 $R$ itself and also $r * Z$ 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 $R$ or $r * Z$ ?

Zachariah Norris 2022-09-15

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

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