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Questions tagged [continuity]

Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)

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Sum of function values

I have no clue how to approach this problem.
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2answers
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Continuity Proof “If $f(x,w)$ is continuous and its domain is a cartesian product, $\max_{w}f(x,w)$ is continuous.”

I posted several questions about the continuity of $max$ functions. Let me assume that the function $f(x,y,z,w)$ is continuous. Is the $\max_{w} f(x,y,z,w)$ continuous? Continuity proof for compact ...
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1answer
33 views

Continuity proof “If $f(x,y,z,w)$ is continuous and domains of $x,y,z,w$ are all convex set, then $\max_{w} f(x,y,z,w)$ is continuous.”

There is the answer about the continuity. Continuity proof for compact domain I want to know how to prove that "If $f(x,y,z,w)$ is continuous and domains of $x,y,z,w$ are all convex set, then $\max_{...
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0answers
25 views

calculate $f(500)=?$ and give an example of a function that meets the given conditions

$f : \mathbb{R} \to \mathbb{R}$ $\forall_{x\in \mathbb{R}} f(x) * f(f(x))=1 (*) \\ f(1000)=999$ calculate $f(500)=?$ So $f(1000)*f(f(1000))=1 \to 999*f(999)=1 \to f(999)= \frac{1}{999}$ Darboux. $\...
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1answer
29 views

Continuity proof for compact domain

I posted the question about continuity, Let me assume that the function $f(x,y,z,w)$ is continuous. Is the $\max_{w} f(x,y,z,w)$ continuous? and got the answer that the function is continuous when ...
5
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1answer
43 views

Let me assume that the function $f(x,y,z,w)$ is continuous. Is the $\max_{w} f(x,y,z,w)$ continuous?

Let me assume that the function $f(x,y,z,w)$ is continuous. Is the $\max_{w} f(x,y,z,w)$ continuous? Since $f(x,y,z,w)$ is continuous, it is seperately continuous for each $x,y,z,$ and $w$.
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1answer
30 views

Calculus Differential Equation

Problem: Let $f$ be a continuous function whose domain includes $[0,1]$, such that $0 \le f(x) \le 1$ for all $x \in [0,1]$, and such that $f(f(x)) = 1$ for all $x \in [0,1]$. Prove that $\int_0^1 f(...
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1answer
25 views

Can continuous functions have removable discontinuities?

I'm trying to resolve what seems like an inconsistency between the epsilon-delta definition of continuity and the limit-based definition ($\lim_{x->c} f(x) = f(c)$). Assume $c$ is a cluster point. ...
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0answers
22 views

Is convex , lower semicontinuous function with closed domain continuous relative to its domain? [duplicate]

Let $f: \mathbb R^n \to [0 , +\infty] $ be a lower semicontinuous and convex function. Assume $\mbox{dom} f = \{ x \in \mathbb R^n | f(x) < +\infty \}$ is a closed set. Prove or disprove that for ...
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2answers
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Sublevel sets of continuous functions are closed

Consider a continuous function $f:\mathbb R^n\to\mathbb R$. The set $$M=\{x\in\mathbb R^n ; f(x)\leq c\}$$ where $c$ is a real number. It follows from continuity that $M$ is closed. I have seen a ...
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1answer
62 views

Does the closest point on a subset change continuously?

$\newcommand{\til}{\tilde}$ Let $(X,d)$ be a metric space, and let $S \subseteq X$. Suppose that every point in $X$ has a unique closest point in $S$, which we denote by $\tilde s(p)$. Is it true ...
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1answer
28 views

Two different conclusions about continuity implying convergence of sequences - are both true?

I used to think the following was true: $f$ is continuous, IFF the following holds: Given any sequence in its domain that converges, a corresponding sequence in its codomain converges. But ...
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0answers
27 views

Uniform Bound on Infinitely Differentialble functions

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Problem Show that if $f$ ...
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2answers
23 views

Understanding proof of sequential continuity?

I'm trying to understand proof of the following statement: Q. Let $f$ be a function on a closed bounded interval $[a,b]$. Prove that $f$ is continuous at $ c \in [a,b]$ if and only if $f(x_n) \to c$...
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0answers
30 views

Suppose that f : [a, b] → R is continuous and that f([a, b]) ⊂ [a, b]. Prove that there exists a point c ∈ [a, b] satisfying f(c) = c. [duplicate]

Suppose that $f : [a, b] \to \mathbb{R}$ is continuous and that $f([a, b]) \subset [a, b]$. Prove that there exists a point $c \in [a, b]$ satisfying $f(c) = c$. (If either $f(a) = a$ or $f(b) = b$ ...
2
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2answers
36 views

continuity of the inverse in $0$ using “the pre image of an open set is open”.

I'm focusing on $\frac 1 {x^2}$, but any function with a "jump" or going to infinity at one point works. I was trying to convince myself about the definition of continuity using the topological ...
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0answers
24 views

Continuity relative to the domain of a convex function.

Let $f: \mathbb R^n \to [0 , +\infty] $ be a lower semicontinuous, convex, and positively homogeneous degree-$2$ function. Assume $\mbox{dom} f = \{ x \in \mathbb R^n | f(x) < +\infty \}$ is a ...
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1answer
23 views

Probably dumb limit

I have a sequence of continuous functions $f_n : I^k \rightarrow I^k$ converging uniformly to a continuous function $f$. Then for each $n$ I choose a point $x_n$ and since they're chosen in $I^n$ ...
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1answer
66 views

Is $f(x)=x$ uniformly continuous on $\mathbb{R}$? [on hold]

Is $f(x)=x$ uniformly continuous on $\mathbb{R}$? I know that $x^2$ is not uniformly continuous on $\mathbb{R}$ but not too sure about $f(x)=x$.
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3answers
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Trouble reading this proof of characterization of continuous mappings by open sets.

Here's the first part of the proof—the part I have a question about. Simply put, I don't see anything here precluding $x_0$ from being on the boundary of $S_0$. And if that's the case then $N_0 \...
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0answers
14 views

Sufficient conditions for continuous-differentiability of a component of a composite function

Suppose $h=g\circ f$. Are there conditions on the functions, $h$ and $f$, that ensure continuous-differentiability of $g$? Clearly, continuous-differentiability of $h$ and $f$ is insufficient. (...
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0answers
22 views

Continuous adjoint of the one-dimensional Laplace equation

Say I have a problem given by the 1D Laplace equation, $$ R (T(\alpha), \alpha) = \frac{d^2 T(x)}{dx^2} - \alpha(x) T (x) = 0, $$ with $x \in [0,1]$, Dirichlet boundary conditions on $x=0$ and $x=1$, ...
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0answers
20 views

Under what conditions are partial derivatives continuous?

For any continuous scalar and vector fields which doesn't contain any singular points, under what conditions are their partial derivatives continuous?
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1answer
45 views

How to prove $\mathbf{M}/r$ is a continuously differentiable vector field?

I had previously asked a question here regarding the applicability of divergence theorem. $\mathbf{M'}$ is a continuous vector field in volume $V'$ (which is compact and has a piecewise smooth ...
3
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2answers
79 views

Continuity of Linear Operator Between Hilbert Spaces

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Let $\mathcal{H}$ be a ...
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votes
1answer
28 views

Function that satisfies no Lipschitz condition on [a,b] but of bounded variation [closed]

I am looking for an instance of a function that satisfies no Lipschitz condition on [a,b] but of bounded variation. I have no idea how to approach this problem.
1
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1answer
22 views

How to find out the zeros of the function?

Let $f : \mathbb R \to \mathbb R$ be a continuous $2\pi$-periodic function, i.e. for every $t \in \mathbb R$, we have $f(t) = f(t+2\pi)$. Prove that there exists $t_0\in \mathbb R$ such that $$f(t_0) =...
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1answer
23 views

Continuity for the composition of 2 functions

Q) Consider the following functions $f(x) = \left\{\begin{matrix} 1, & |x| \leq 1 \\ 0, & |x| > 1 \end{matrix}\right.$ and $g(x) = \left\{\begin{matrix} 1, & |x| \leq 2 \\ 2, &...
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1answer
24 views

If f is Lebesgue integrable, then $g(x) = \int_x^{\infty} f(y) d \lambda (y)$ is continuous.

This is a true or false question, $f,g$ both $ \mathbb{R} \rightarrow \mathbb{R}$. I think it is right, but in the proof, I have to show for any positive $\epsilon$, there exists positive $\delta$, ...
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0answers
41 views

Continuous Functions on $\ell^2$

let ${a_n}\in\ell^2$ . prove that $f$ defined by $f({b_n})=\sum_{n=1}^{\infty}a_nb_n$ is a continuous real-valued function on $\ell^2$
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1answer
64 views

Construct a continuous function $g$ which vanishes on closed $F$ and “follows” a continuous function $f$ outside $F$.

Let $X$ be a compact Hausdorff space and $F$ be a closed subspace. Let $f\colon X \to \mathbb R$ be a continuous function. I want to construct a continuous function $g \colon X \to \mathbb R$ such ...
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1answer
27 views

Utility function and preference relations

If a utility function represents a consumers preference relation if it assigns 'higher numbers' to preferred bundles, how do we know the properties of this preference relation (i.e. complete, ...
3
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1answer
27 views

Redundant definition of computable function on $[a,b]$ in Pour-El and Richard book?

In Pour-El and Richard Computability in Analysis and Physics page $25,$ they defined computable function on $[a,b]$ as follows: Let $[a,b]$ be such that $a$ and $b$ are computable real numbers. A ...
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3answers
49 views

Check the continuity of a function

Given $$f(x) = \begin{cases} 0, & x \in [-2,2]\\ x-2, & x > 2\\ x+2, & x < -2, \end{cases} $$ check whether $f$ is continous on $\mathbb{R}$. I used one-side limits in order ...
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1answer
38 views

Steps to determine the interval of continuity of $f(x)= \sum_{n=2}^{\infty} \frac{(\sin{nx})^2}{\sqrt{n}}$

I am trying I determine the interval of continuity of $$f_n(x)= \sum_{n=2}^{\infty} \frac{(\sin{nx})^2}{\sqrt{n}}$$ I tried to find the domain by Dirichlet's test, but the sum $\sum _{n=1}^{\infty }(\...
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2answers
147 views

Continuous function with $f(x^m+y^n) \le f(x+y) $

Let $m>n\ge 1$ be integers. If $m$ is even and $f:\mathbb{R} \to \mathbb{R} $ is continuous, nonconstant, with $f(x^m+y^n) \le f(x+y) $, $\forall x, y \in \mathbb{R} $, prove that $n$ is even. I ...
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2answers
37 views

If $K$ is compact and $f\colon K\to\Bbb R$ has the intermediate value property, does $f$ attain its extrema?

Assume that $K\subset \Bbb R$ is compact and $f\colon K\to\Bbb R$ is a Darboux function, i.e. it has the intermediate value property. Set $$ m:=\inf_K f, \quad M:=\sup_K f.$$ As discussed here, $(m,M)...
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2answers
41 views

Question about the proof of the $\epsilon-\delta$ definition of continuity

I am currently trying to get my head around the proof of the definition of continuity of a function given in my Elementary Analysis textbook. The definition given is: Let $f$ be a real-valued function ...
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1answer
49 views

Continuity on infinity

We have $f: [a.b] \rightarrow \mathbb{R}$ continuous, and $$c_n = \sup\{c \in [a,b] : |f(x) - f(c_{n-1})| < \epsilon \text{ for any } x \in [c_{n-1},c]\}$$ with $c_1 = a$. In a previous exercise ...
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1answer
56 views

What does it mean for partial derivative to exist but not continuous?

What does it mean for partial derivative to exist but not continuous? Continuous partial derivatives imply differentiability and differentiability imply continuity of a function and the existence of ...
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0answers
55 views

Why is it necessary to show subsequence convergence in the extreme value theorem?

I’m probably making this more complicated than it needs to be, but I’m trying to figure out why it is necessary to prove the convergence of a subsequence in proving the extreme value theorem (EVT). I ...
4
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1answer
92 views

If $f(x) + f(2x)$ is continuous, is $f$ continuous or not?

True or false: If $g(x)=f(x)+f(2x)$ with $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, then $f$ is continuous. My idea was to find a counterexemple since, first, I claim that this is false. ...
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1answer
40 views

Find an increasing continuous function such that : $\mid p(x) - p(y) \mid \leq w(\mid x - y \mid)$

Let $p : [0,1] \to \mathbb{R}$ be a continuous function. Prove the existence of an increasing and continuous function $w : [0,1] \to \mathbb{R}$ such that : $\lim_{x \to 0} w(x) = 0$ and such that : ...
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1answer
29 views

proof lim x-a f(x)= lim x-0 f(x+a) ( duplicate)

i guess the proof here(Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$) is something wrong, its too easy and i thought i couldnt change things like the one most voted
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2answers
25 views

Bounded metric in compact metric space with continuous function

Let $X$ be a compact metric space with metric $d$ and $f:X \to X$ a continuous map so that $f(x)$ never equals $x$. How do I show that the function $g(x) :=d(f(x),x)$ is continuous?
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2answers
54 views

Calculus tips question [closed]

For what values of $c$ is the function: $$f(x)= \begin{cases}(cx-1)^3 &\text{ if } x>2\\ c^2x^2-1& \text{if } x\le 2 \end{cases}$$ continuous at every number? Can ...
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0answers
44 views

Is $f(t) = e^{X(t)}$ continuous when matrix $X(t)$ is continuous?

I think $f(t) = e^{X(t)}$ is continuous over $t \in \mathbb{R}$ when the complex matrix valued mapping $t\mapsto X(t)$ is continuous - regardless of whether $X$ is infinite or finite dimension. I ...
2
votes
1answer
28 views

How to show locally Lipschitz on $A \subset \mathbb{R}^n$ implies continuity on $A$?

I know this question has been answered before but the way that I am trying to show is different and uses the open balls for the proof. I need a clarification for some part of my proof. Let $A \subset ...
0
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0answers
33 views

Proof of continuity of a vector function

Let $f: \mathbb{R}^m \rightarrow \mathbb{R}$. Show that $f(x) = x_i$ where $x_i$ is the $i^{nd}$ component of $x$ regarding the standard basis of $\mathbb{R}^m$ is continuous in 0. The definition ...
1
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2answers
44 views

Prove that a specific continuous function is bounded

Let $f:$ $\mathbb{R}$ $\to$ $\mathbb{R}$ be a continuous function such that $f(x +1) = f(x)$ for all $x \in \mathbb{R}$. Show that $f$ is bounded. My first thought is that $f(x)$ is a constant ...