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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Continuity of $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^2 - \{ (0,0) \}$ using epsilon and delta definition

I recently tried to prove the continuity of this function $\theta(x,y) = \frac{(x,y)}{\sqrt{x^{2} + y^{2}}}$ in $\mathbb R^{2} - \{ (0,0) \}$ using the inverse image of some open in $\mathbb R^2$ but ...
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How to prove that the convergence of $(f_n)$ is uniform on compact sets?

I'm reading Theorem 0.9 in this lecture note. Below is my attempt where I got stuck at the end. Could you elaborate on how to finish the proof? Let $C$ be an open convex subset of a Banach space $X$....
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Showing that $f((x_1,x_2),(y_1,y_2)):=x_1-y_1$ is continuous.

Define $f:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$ by $$ f((x_1,x_2),(y_1,y_2)):=x_1-y_1 $$ I want to show that $f$ is continuous. I already know that the function $g(x,y):=x-y$ is ...
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Proof of Bloch's theorem: continuity of $h(r)$

Bloch's Theorem. Let $f$ be an analytic function on a region containing the closure of the disk $D= \lbrace z ; |z|<1 \rbrace$ and satisfying $f(0)=0$ and $f'(0)=1$, then there is a disk $S\subset ...
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Is this function arising from measure of a certain set continuous?

Let $F$ be a CDF of random variable $X$ distributed in $[0,1]$. $F$ has a density, $f$. Let $\mu_F$ denote the corresponding measure, i.e. $\mu_F(A)=\int\limits_{A}f(x)dx$ for all $X \in \mathcal{B}([...
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If $\mathcal{F}$ is pointwise bounded, then $\mathcal{F}$ is locally equi-Lipschitz and locally equi-bounded

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X$ be a Banach space and $\mathcal{...
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Can a continuous function $f$ with no flat regions have infinite solutions for $f(x)=c$?

Let $f:[0,1]\rightarrow[a,b]$, $f(\cdot)$ is continuous and has no flat regions. Can the equation $f(x)=c$ have an infinite number of solutions? It seems pathological cases like the Weierstrass ...
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Uniform convergence on [a, b] of series of continuous functions implies that series is continuous on (0, b]

(1) Let's say we have a series $f = \sum_{n=1}^\infty f_n$ which converges uniformly on every closed interval $[a, b]$ where $a > 0$ is arbitrary and $b$ is fixed. Moreover $f_n$ is continuous for ...
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Why do we use H in the limit here? And how do you cancel out |h|/h, is that even allowed?

This is the original function: $f_3(x, y) = |x|e^{x^2y}$ This is the limit we used: \begin{align*} \lim_{h \to 0^+} \frac{f_3(h, y) - f_3(0, y)}{h} & = \lim_{h \to 0^+} \frac{|h|e^{h^2y}}{h} = \...
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Is $f :\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}\times\mathbb{R},\ f((x_1,x_2),(y_1,y_2)):=(x_1,y_1)$ continuous?

Consider the real line $\mathbb{R}$ with the topology that has a basis consisting of open intervals, and equip $\mathbb{R}^2=\mathbb{R}\times\mathbb{R}$ and $\mathbb{R}^2\times\mathbb{R}^2$ with the ...
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Example of discontinuous convex l.s.c. function on an open convex subset of an incomplete normed space

I'm reading Proposition 0.7. in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is continuous on $C$. (b) If ...
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Does my proof generalize this exercise about continuity of convex lower semi-continuous function?

I'm trying to solve below question (Proposition 0.7.) in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is ...
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Question regarding proof that multiplication of equivalence classes of paths is associative.

How do you show that $F$ as defined above is continuous? I do understand intuitively why $F$ is defined the way it is and I can understand intuitively why $F$ is continuous. But I'm not sure how you ...
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Continous function a on convex set [closed]

Let $f:\Omega\rightarrow \mathbb R$ be a continous function where $\Omega$ is convex set. Does that mean that $f(\Omega)$ also a convex set?
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In a normed vector space, the convex function $f:C \to \mathbb R$ is locally Lipschitz if and only if $f$ is upper bounded on an open subset of $C$

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $(X, \| \cdot\|)$ be a normed ...
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Showing a contradiction involving an open set restriction and a continuous function

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that for all $x \in U_X$ we have $f(x) \leq 0$ and for all $y \in U_Y$ we have $f(y) > 0$, where $U_X \cap U_Y = \emptyset$....
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Is the map continuous?

Consider two topological spaces: The set of positive integers $\mathbb{N}$ and the topology $\mathscr{M}$ given by $\emptyset$ and the subsets of $\mathbb{N}$ that contain $1\in\mathbb{N}$. The set $[...
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What is range of values that the word 'nearby' supposed to represent in this informal definition of continuity.

In my book it gave two informal explanation for the concept of continuity. I had doubt in the second explanation but I cleared it by asking it here. The explanation is , Suppose a function f has the ...
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In this informal definition of continuity (at $x=p$), what does "regardless of the manner in which $x$ approaches $p$" mean?

In my book it gave the informal definition of continuity as If we let $x$ move toward $p$, we want the corresponding function values $f(x)$ to become arbitrarily close to $f(p)$, regardless of the ...
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Find a discontinuous linear transformation from the set of bounded real sequences to the reals

I am looking for a linear transformation T: B(N,R) -> R, with R the real numbers and B(N,R) the set of all bounded real sequences (with the sup norm), that is discontinuous. I first thought that ...
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What does "contained continuously" mean? As in, "metric space $X$ is contained continuously in metric space $Y$".

What we mean by the "Contained continuously" terminology?! Definition: Let $H$ be a Hilbert space and $E$ is an open bounded subset of $\mathbb[C]$. The Hilbert space of measurable ...
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Prove the $\sin$ function is continuous at $0\in\mathbb{R}$

I need to test this and I don't understand how the first part is related to the second part. Claim Use the identity $|\sin(x)| \leq |x|$ when $0 < |x| < \pi/2$ to show that $\sin(x)$ is ...
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4 votes
3 answers
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Proof: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$?

Question: How many continuous/bounded functions on $[0,1]$ verify $f(x)=f(x/2)\frac{1}{\sqrt{2}}$? Answer: Thank to @TonyK @Ryszard Szwarc. I think that i found an ever stronger demonstration that ...
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Show that $M \setminus f^{-1}(c)$ is disconnected if $f:M\rightarrow \mathbb{R}$ is continuous and $c \in (\min f(M), \max f(M))$.

Let $f : M \rightarrow \mathbb{R}$ be a continuous function. Prove that if $c \in \mathbb{R}$ is strictly between minimum and maximum of $f$ in $M$ then $M \setminus f^{-1}(c)$ is disconnected. Can ...
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Is $\Lambda:X\longrightarrow\mathbb K$ continuous if $\{\Lambda(x_k)\}_{k\in\mathbb N}$ is bounded when $x_k \longrightarrow0$?

Let $X$ a normed space over $\mathbb K$ ($=\mathbb R$ or $\mathbb C$) and $\Lambda:X\longrightarrow\mathbb K$ a linear aplication. Prove that if $\{\Lambda(x_k)\}_{k\in\mathbb N}$ is bounded $\forall(...
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2 answers
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How to find the primitive/ antiderivative of a discontinuous function

I have the following function before me: $$f(x)=\left\{\begin{array}{ll}\phantom{-}x^2, &x\leq 0\\-x^2+2,&x>0\end{array} \right .$$ I have to find a function $g(x)$ such that $g'(x)=f(x)$. ...
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$f''(x)$ is always positive then $f(x+f'(x)) \geq f(x) $ [duplicate]

$f: R \to R$ be such that $f''(x) >0$. Prove that $f(x+f'(x)) \geq f(x) $. My thought: $f''(x) >0$ means f is concave up but $f'(x) $ can be either positive or negative or may be mixed (positive ...
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Exercise on continuous functions, analysis 1

a) Let $f$ be a continuous function on $[a, b]$. Show that if $f(r) = 0$ for every rational number $r\in [a,b]$, then $f(x) = 0$ for all $x\in[a, b]$. b) Let $f$ and $g$ be continuous functions on $[a,...
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2 answers
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Must a certain continuous map have 0 in its image, given that its restriction to the unit sphere is homotopic to the identity?

Suppose $f:\mathbb{B}^n \to \mathbb{R}^n$ is continuous (here $\mathbb{B}^n$ refers to the $n$-dimensional unit ball). Suppose also that its restriction $g := f|_{\mathbb{S}^{n-1}}$ does not have $0$ ...
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1 answer
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Continuity of Taylor remainder for a multivariate $C^1$ function

Suppose I have a function jointly $C^1$ in two variables. By Taylor's theorem I can expand in one of the variables, say the second, and get a remainder term $R(x,y,h) = \frac{f(x,y+h) - f(x,y)}{h} - \...
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Is this convolution continuous?

Let $0 \leq \xi \leq T$, $Y_0 \in \mathcal C([0, 2T])$, $a_1 \in L^2([0,T])$ and consider the following function: $$ Y_{1}(\xi)=\int_{0}^{\xi} a_{1}(\alpha) Y_{0}(\alpha+\xi) d \alpha $$ Now if inside ...
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4 votes
2 answers
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Continuity of polar decomposition

This question is about a step in the proof of this answer, which is not directly clear to me. Consider the following scenario: $H$ is a Hilbert space, $A\in GL(H)$ a positive self-adjoint operator. We ...
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Continuity a function of several real variables

The goal is to study the continuity of the following function: $f(x, y)= \begin{cases}x^{2}y, & \text { si }|x|<y \\ y, & \text { otherwise. }\end{cases}$ $D_{f}=\left\{(x, y) \in \mathbb{R}...
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Can someone explain this proof of continuity of that the function of several real variables? [closed]

The goal is to study the continuity of the following function: $f(x, y)= \begin{cases}x^{2}, & \text { si }|x|<y \\ y^{2}, & \text { otherwise. }\end{cases}$ $D_{f}=\left\{(x, y) \in \...
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1 answer
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Lipschitz continuous function

I have this function and I would like to know if it is Lipscitz continous but I don't know how and from where I could start. $$m \frac{dv}{dt} = F_e + F_h - mg \; sin\alpha - C_{rr}mg \; cos\alpha - ...
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f(x) is defined for the interval [a, b] and continuous, where f(a) = f(b) = 5 and b - a = 24, show that there exist x such that f(x) = f(x + c). [duplicate]

Is this possible for any c, 0 < c < 24? If not, then in which case is it possible and how to show either of the cases with Intermediate Value Theorem?
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f(x) is defined for the interval [a, b] and continuous, where f(a) = f(b) = 5 and b - a = 24, show that there exist x such that f(x) = f(x + 12). [duplicate]

I know that this question can be solved using Intermediate Value Theorem, but I don't know how to show that such a point exist between the interval. Ideally, it would be very helpful to show that ...
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1 vote
1 answer
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Determine if the functions are differentiable at $x=4$

analyze the function $$f(x)= \begin{cases}\sqrt{4x}+11 & \text{if }x \ge 4 \\ \frac12 x+13 & \text{if }x < 4 \end{cases}$$ I am trying to determine if the equations are differentiable at $x=...
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2 answers
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Limit of the continuous function [closed]

Simplify the subsequent function: $f(x)=\lim\limits_{n \to \infty}\frac{ln(e^n+x^n)}{n}$ and $n>0, x>0$ Already known: The function is continuous in $\mathbb{R}$. And when $x\leq e$, the ...
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6 votes
2 answers
182 views

$\int _0^1f^2\left(x\right)dx-2\int _0^{\sqrt{3}-1}\:\left(x+1\right)f\left(x\right)dx\:+1=0$

Let $ f $ be an increasingly continuous function over $[0,1]$ such that: $$\int _0^1(f\left(x\right))^2dx-2\int _0^{\sqrt{3}-1}\:\left(x+1\right)f\left(x\right)dx\:+1=0$$ Find all functions $f$ ...
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Proving the principal argument not continuous using standard metrics [closed]

Let $\operatorname{Arg}: \Bbb{C} \setminus \{0\} \to\Bbb{R}$ be the principal value of the argument, taking values in $(−\pi, \pi]$. Using the standard metrics on $\Bbb{C} \setminus \{0\}$ and $\Bbb{R}...
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Regarding proof of Bolzano's theorem (Csez Kosniowski)

I am trying to understand the lemma 10.1 (IVT) of "A first course in Algebraic topology" by C. Kosniowski. The lemma states, If $f: I \rightarrow \mathbb R$ continuous with $f(0)f(1) \leq0$, ...
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Let $m,n\in\mathbb{Z}^+$ & $n$ odd. Let $f:\mathbb{R}→\mathbb{R}$ by $f(x)=(x^{1/n})^m=x^{m/n}\quad\forall x$. Show $f$ is continuous on $\mathbb{R}$ [closed]

Problem Let $m$ and $n$ be positive integers and $n$ is odd. Define the function $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=(x^{1/n})^m=x^{m/n}\quad\forall x$. Show that $f$ is continuous on $\...
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2 votes
4 answers
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Is there any epsilon delta proof, in which we use the bare minimum of $\delta $ existing?

In every epsilon delta proof I see, it ends up with reverse engineering the epsilon inequality to find a suitable delta. Ultimately, it seems so that one can always write down $\delta$ as a function ...
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2 answers
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Epsilon-delta definition in proving the continuity of $\frac{1}{2}x^2$

How can I prove that the function $$ f:\mathbb R\rightarrow \mathbb R$$ $$ x\mapsto\frac{1}{2}x^2$$ is continuous? I currently have: $$ \epsilon > 0, \delta > 0$$ $$ |x-a| < \delta \...
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2 answers
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Measure theory showing integral of non-negative function is continuous

Let $f:(\mathbb{R}, B(\mathbb{R})\rightarrow(\mathbb{R},B(\mathbb{R}))$ be a non-negative function and $\int_{\mathbb{R}}f d \lambda < \infty$. $F:\mathbb{R} \rightarrow \mathbb{R}, F(x):= \int_{(- ...
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-2 votes
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Does finiteness of the definite integral over a function imply continuity? [closed]

For example, if \begin{align*} \int_{a}^{b} f(x)\mathrm{d}x = C \end{align*} for some finite real number $C$, is $f:[a,b]\rightarrow\mathbb{R}$ continuous on $[a,b]$ ? If not, is there any further ...
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0 votes
1 answer
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Ideals of $C^0([0, 1]; \Bbb R)$ and compactness [duplicate]

Let $C := C^0([0, 1]; \Bbb R)$ the ring of continuous real functions on $[0, 1]$. Let $I \subset C$ an ideal. We suppose that $I$ is not contained in any $I_x:= \{f \in C \lvert f(x) = 0\}$. Show that ...
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Existence of a function $f:\mathbb{R}^8\to \mathbb{R}$ [closed]

Write $(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)$ for points in $\mathbb{R}^8$. Determine if there exist a function $f:\mathbb{R}^8\to \mathbb{R}$ satisfying all of the following: $f$ is continuous and ...
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1 answer
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Does this implies $(X, \tau_X) $ discrete space?

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological space such that $card(\tau_Y)\ge 3$ $f\in C(X, Y) \quad , \forall f \in Y^X$. Does this implies $(X, \tau_X) $ discrete space? I know if the ...
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