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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Pointwise convergence of sequence of continuous functions to a continuous function

Let $\{f_n\}$ be a sequence of continuous functions from $\mathbb{R}$ to $\mathbb{R}$, and $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. Suppose that $\{f_n\}$ converges to $f$ at ...
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Prove or disprove $C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}$ is closed

Let $f:\mathbb{R}^n\to\mathbb{R}$. Given $x_0\in\mathbb{R}^n$, define $$C:=\{x\in\mathbb{R}^n|f(x)\leq f(x_0)\}.$$ Show that $C$ is closed. My attempt: Let $\{x_n\}$ be a sequence in $C$ that ...
lee max's user avatar
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0 answers
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What can we say about $f$ and $g$?

Suppose $f$ and $g$ are holomorphic on a bounded domain $D$ and continuous on $\bar D$. Suppose also $|f(z)|=|g(z)|\neq0$ on $\partial D$ and $\frac{|g(z)|}{3}\leq|f(z)|\leq 3|g(z)|$ for all $z\in D$. ...
Derewsnanu's user avatar
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0 answers
41 views

Calculating Probability for Absolute Value of Continuous Random Variable within Interval [closed]

I'm working on a probability problem involving a continuous random variable X. Since the interval involves the absolute value of X, I'm uncertain about how to approach this calculation. Could someone ...
ismailmushraf's user avatar
1 vote
1 answer
44 views

Question on proving continuity in Real analysis, using epsilon-delta definition

My lecturer did this example: Show, using the definition, that $f : \mathbb{R → R}$ given by $f(x) = x^2$ is continuous at $a = 3.$ This was the model solution: Let $\epsilon ∈ \mathbb{R}+.$ We first ...
Tay's user avatar
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Uniform convergence and continuous limit in $C^0$ and $L^{\infty}$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be $2\pi$-periodic and $f_n \in C^0_{2\pi}(\mathbb{R})$ for all $n \in \mathbb{N}$ such that $\lim_{n \rightarrow \infty} \| f_n-f\|_{\infty} =0$. Then $f \...
Oscar210899's user avatar
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Show that the jump $j_f(c)$ of increasing $f$ at $c$ is given by $\inf\{f(y)-f (x): x < c < y, x, y \in I\}$.

Let $I\subseteq \mathbb{R}$ be an interval and let $f: I \to \mathbb{R}$ be increasing on $I$. If $c$ is not an endpoint of $I$, show that the jump $j_f(c)$ of $f$ at $c$ is given by $\inf\{f(y)-f (x):...
user13's user avatar
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When does $F(F^{-1}(p))=p$ fail, where $F$ is a CDF?

We define the inverse of a CDF $F$ as $$F^{-1}(p)=\inf\{x:F(x)\ge p\},\quad p\in(0,1).$$ It is easy to see that $F(F^{-1}(p))\ge p$ since $F^{-1}(p)$ satisfies the condition that $F(x)\ge p$. I am ...
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continuity for limit of pointwise convergent functions

We Know that "The uniform limit of Continuous Functions is continuous." and here is its proof: We want to show that $f$ is continuous at a point $x_0$, say. The condition for continuity says ...
A12345's user avatar
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2 votes
1 answer
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Continuity of a mapping on a space of polynomial

I consider the set $$ \mathcal{U}=\{u_{a,b}:x\in[0,1]\mapsto ax^2+bx : a\leq 0, [b\leq a\quad\text{or}\quad b\geq -a]\} $$ And the set of associated coefficients $$ \mathcal{C} =\{(a,b)\in\mathbb{R}^{...
coboy's user avatar
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How do I find $a$ and $b$? [closed]

Find $a$ and $b$, so that $f(x)=\begin{cases}(a+1)e^{x-1}, &x\le1;\\ \sin(x-1)+b\cos(2x-2), &x>1\end{cases}$ is differentiable. I calculated left hand limit and right hand limit, and got ...
allee011's user avatar
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A question about continuous functions with compact support in locally compact topological groups.

I'm having difficulties proving the following statement: If $G$ is a locally compact Hausdorff topological group and $C_c(G)$ is the space of continuous real functions on $G$ with compact support, ...
Matheus Frota's user avatar
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1 answer
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Subset of $C^1([0 , 1])$ is open

Let $C^1([0 , 1],\mathbb{R})$ be the subspace of $C([0 , 1],\mathbb{R})$ of the functions that have a continuous derivative throughout $[0 , 1]$ with norm: $\|f\|=\sup_{x \in [0,1]} \{ |f(x)|+|f'(x)| \...
evelyn juarez's user avatar
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Show that the level set of the sum of a continuous and a semicontinuous function is closed.

Let $$E_{\alpha} = \{z \in \mathbb{R} : zt - \log(\varphi(t)) \leq \alpha\}$$ be the level sets of $Q_t(z) = zt - \log(\varphi(t))$, $t \in \mathbb{R}$, with $\alpha \in [0, \infty)$. I`ve already ...
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1 answer
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Finding the discontinuities of $\text{inf}_i (|x - q_i| 2^i)$

Let $q_i$ be an enumeration of the rational numbers, and let $f(x) = \text{infimum}_i (|x - q_i| 2^i)$ obviously f(x) is 0 and continuous at every rational number. Additionally, f is discontinuous at ...
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Points of discontinuity of greatest Integer function

So here's my problem Find the points of discontinuity of $[x^2]$ in the interval$[-1,2]$ where [] is the G.I.F. I am not able to understand why at $2^{1/2}$ and $3^{1/2}$ the function is being ...
πααρτθ Σαρθι's user avatar
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1 answer
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Topological continuity of a circle parametrization

A canonical example of a map between topological spaces that is continuous and bijective, but whose inverse is not continuous (thus preventing it from being a homeomorphism) is a parametrisation of a ...
serpens's user avatar
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Continuity on countably-normed Hilbert spaces

i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an ...
Marco Lugarà's user avatar
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The derivative of a function of several variables

Define a function $f:\mathbb{R}^2\to\mathbb{R}$ by: $$f(x,y)=\begin{cases} \dfrac{x\sin y - y\sin x}{x^2+y^2}\ &\text{if}\ (x,y)\neq(0,0)\\0\ &\text{if}\ (x,y)=(0,0) \end{cases}.$$ I want to ...
Lê Trung Kiên's user avatar
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2 answers
65 views

Proving $f$ is the Identity Function Given $f^n(x) = x$ for All $x \in [0,1]$ [duplicate]

Let $f : [0,1]\rightarrow[0,1]$ be a continuous function with $f(0)=0$. Let $f^n = f \circ f \circ ... \circ f$ (composition n times). Proof that if $f^n(x)=x$ for all $x \in [0,1]$ for some fixed $n \...
mick87's user avatar
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1 answer
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Is this Result on Continuity of Composite functions true?

Consider, A real valued function $f(x)$ which is discontinuous $\hspace{2mm}\forall \hspace{2mm} x \in [\alpha,\beta]$ and another real valued function $g(x)$ which is discontinuous $\hspace{2mm}\...
Jesko's user avatar
  • 27
1 vote
0 answers
71 views

Is there any other way to make a $f :[0,1]\to \mathbb{R}$ discontinuous at every point of its domain?

The Dirichlet function: $$f(x) = \begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational } \end{cases}$$ is an example of a function that is discontinuous everywhere ...
pie's user avatar
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2 votes
1 answer
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"No infinite discrete subspace" vs "No infinite pairwise disjoint family of opens"

What is the relationship between the two properties (for a topological space $X$) $A$: "$X$ has no infinite family of pairwise disjoint open subsets" $B$: "$X$ has no infinite discrete ...
Chris Grossack's user avatar
6 votes
3 answers
133 views

If $f$ is continuous at $a$, is $f^{-1}$ continuous at $f(a)$?

Let $I\subseteq\mathbb{R}$ be an open interval, and $f:I\to\mathbb{R}$ an injective function. Let $a\in I$, and suppose that $f$ is continuous at $a$. Does it follow that $f^{-1}$ is continuous at $f(...
ashpool's user avatar
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Hypothesis check: The graph of a bounded function on $(0, 1)$ is not closed

Consider the following problem: Let $f: (0,1) \rightarrow \mathbb{R}$ be a continuous and bounded function. Prove that the graph $\Gamma(f)$ of $f$, $$ \Gamma(f) = \left\lbrace (x, f(x)) \mid x\in (0,...
Iván G M's user avatar
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0 answers
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Periodic Bell Curve?

I'm trying to code a hue selection function in a circular color model (in which each hue is attributed a value between 0 and 1) so I'm working on a Gaussian function and would like it to roundtrip ...
Raphael Jaafari's user avatar
3 votes
2 answers
100 views

Discontinuous solution of $y''(x)-2(1-x)(y'(x))^2=0$ with $y(0)=1$ and $y(2)=-1$

I tried to solve the equation $$y''(x)-2(1-x)(y'(x))^2=0$$ with the conditions $$y(0)=1, y(2)=-1.$$ It's easy to verify that the function $$y(x)=\frac{1}{1-x}$$ satisfies both the equation and the ...
Mohamed Mostafa's user avatar
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0 answers
39 views

Question about proof of Lindelöf Theorem

Supose that $\gamma : [0,1] \to \overline{\mathbb{D}}$ is continuous, $\gamma(t) \in \mathbb{D}$ for $0 \le t < 1$ and $\gamma(1) = 1$. Suppose that $f \in H(\mathbb{D})$ is bounded. If $f(\gamma(t)...
MathLearner's user avatar
0 votes
1 answer
18 views

Continuous Extension of $f : D \rightarrow \mathbb{R}$, $x \mapsto \frac{{x_1 \sin(x_2) + x_2 \sin(x_1)}}{{\sqrt{x_1^2 + x_2^2}}}$

Investigate at which points on $\partial D$ the function can be continuously extended, and in this case, provide the continuous extension. $f : D \rightarrow \mathbb{R}$, $x \mapsto \frac{{x_1 \sin(...
j.primus's user avatar
1 vote
2 answers
82 views

Discountinous function has a zero.

Let $f,g: [a,b] \to \mathbb{R}$ where $g$ is continous function, $f+g$ is non-decreasing and $f(a)>0>f(b)$. Prove that we can find a point $c$ such that $f(c)=0$. Given that $f+g$ is non-...
user avatar
3 votes
1 answer
56 views

Is $\inf\left\{t\in\left[0,1\right]\vert t+B^2_t=1\right\}$ a stopping time?

Problem Let $\left(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P}\right)$ be a filtered probability space such that $(\mathcal{F}_t)_{t\geq 0}$ is a complete and right-continuous filtration ...
Wilfred Montoya's user avatar
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0 answers
14 views

prove existence, uniqueness and non-negativity of the system of ODE

I have an epidemic model with 9 compartments, where all epidemic parameters are non-negative. \begin{equation} \label{model} \begin{split} &\frac{dS}{dt} = \beta N -\delta S-\phi \frac {SI}{...
HRAUNMAKASH's user avatar
0 votes
1 answer
38 views

Continuity and Product Spaces

I have the following basic question regarding continuous functions. Is the following statement correct? Fix nonempty toplogical spaces $A, B, C, D$ with $A \times C$ and $B \times D$ endowed with the ...
Kolmin's user avatar
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2 votes
1 answer
86 views

How is the set $C(f)\cap V$ of second category in $V$?

I am reading the paper "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the ...
Ghosh Da's user avatar
0 votes
1 answer
10 views

A clarification regarding the definition of uniform continuity of a function defined in a subset of $\mathbb R.$

Let $f:A\to \Bbb R$ where $A\subseteq \Bbb R$. We say that, $f$ is uniformly continuous on $A$ if for any $\epsilon\gt 0$ there exists $\delta(\epsilon)=\delta\gt 0$ such that for any $x_1,x_2\in A$ ...
Thomas Finley's user avatar
1 vote
0 answers
22 views

Can a non-constant continuous function be constant on these hyperbolas?

Can a non-constant continuous function $f:\mathbb{R}^2\to\mathbb{R}$ be constant on the following hyperbolas? $$H_a=\{(x,y)\in\mathbb{R}^2:x+1/y=a\},a\in\mathbb{R}$$ $$H_\infty=\mathbb{R}\times\{0\}$$ ...
tripaloski's user avatar
0 votes
2 answers
73 views

The limit of $f(x) = \frac{1}{x^2 + 5x - 24}$ at $x=4$

I'm working through Advanced Calculus: Theory and Practice by John S. Petrovic and is currently working on problem 3.4.2, which is as follows: Find the limit and prove that the result is correct ...
Adam Nygren's user avatar
-2 votes
0 answers
51 views

Prove using $\epsilon$-$\delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$ [duplicate]

Question: Prove using $\epsilon$-$\delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$ Rem: I know that there is a similar question ...
OffHakhol's user avatar
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1 vote
1 answer
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Prove that if $\lim_{x\to c}f'(x)$ exists, the value is $\lim_{x\to c} {f(x)-f(c)\over x-c}$ when $f$ is continuous. [duplicate]

Because $f$ is continuous, by MVT, there is a number d such that $${f(x)-f(c)\over x-c}=f'(d)$$ in (c, x) or (x, c). And if $x\to c$, trivially $d\to c$. So, for $d$ s that have a coresponding x, $f'(...
Zjjorsia's user avatar
3 votes
0 answers
59 views

Prove $\sin|x^2+y|$ is not continuously differentiable [closed]

Let $F:\mathbb{R}^2\to\mathbb{R}, F(x,y)=\sin |x^2+y|$. a) Prove that $F(x,y)$ is not continuously differentiable at $(0,0)$. b) Is $F$ differentiable at $(0,0)$? I can do part b): If $F$ is ...
lee max's user avatar
  • 436
4 votes
1 answer
90 views

Can we make this subspace $\aleph_0$-dimensional?

Let $X$ be a compact Hausdorff space and $A\subseteq X$ a subspace of $X$. Is it possible for the space $\{f\vert_A:f\in\mathcal{C}(X,\mathbb{R})\}$ to be $\aleph_0$-dimensional as a $\mathbb{R}$-...
tripaloski's user avatar
3 votes
1 answer
56 views

If $f$ is Lipschitz and $g \in C^\infty_c(\mathbb{R})$, is $g \circ f$ Lipschitz? [closed]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be Lipschitz continuous and $g \in C^\infty_c(\mathbb{R})$ (i.e. smooth with compact support). Is the composition $g \circ f$ Lipschitz? I have tried proving ...
CBBAM's user avatar
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0 votes
0 answers
14 views

Multi valued function and lower semi continuity

I consider $X$ a metric space and $F_1,F_2$ two disjoint subsets of $X$. Let $T : X\rightrightarrows\mathbb{R}$ be a multi valued function defined by : $T(x) =\{0\}$ on $F_1$ $T(x)=\{1\}$ on $F_2$ $T(...
coboy's user avatar
  • 1,361
1 vote
2 answers
60 views

If $f(0)=0,f(1)=1$,find all $a$ such that $\exists \xi\in (0,1)$ such that $f(\xi)+a=f'(\xi)$

Let V be the set of all contimuous functions $f : [0,1] \to \mathbb{R} $, differentiable on $(0,1)$,with the property that $f(0) = 0$ and $f(1)= 1$. Determine all $a\in \mathbb{R}$ such that for every ...
kmxzc's user avatar
  • 21
0 votes
0 answers
8 views

Continuity of a quantity in a conical system to determine the velocity field

My research is on radar images and the images are collected in several conical surfaces. These conical surfaces have the same origin, the same maximum length (max flare or max range), but different ...
CfourPiO's user avatar
0 votes
2 answers
32 views

Differentiability of a Dirichlet Function Modified with $x^2$

I am quite stumped on a homework question for my real analysis course. The question is as follows: Prove that $g(x)=\begin{cases} x^2 & x\in \mathbb{Q} \\ 0 & x\not\in \mathbb{Q} \\ \end{...
Rob S.'s user avatar
  • 13
2 votes
0 answers
41 views

why is $f(x,y)=\begin{cases}\frac{4(x^2-y)(2y-x^2)}{y^2} & y>0\\ 0 & y\leq 0.\end{cases}$ continuous on $\mathbb{R}^2\setminus\{(0,0)\}$?

I am reading a real analysis book which states that the function $f:\mathbb{R}^2\to\mathbb{R}$ defined by $$f(x,y)=\begin{cases} \frac{4(x^2-y)(2y-x^2)}{y^2} & \text{if } y>0\\ 0 & \...
lorenzo's user avatar
  • 4,042
0 votes
0 answers
36 views

Prove the derivative of order 2 is discontinuous

Let $$f(x,y)=\begin{cases} y^2\ln\left(1+\dfrac{x^2}{y^2}\right)\text{ if }y\neq 0\\ 0\hspace{2.7cm}\text{ if }y=0.\end{cases}$$ Prove that $\dfrac{\partial^2f}{\partial x\partial y}$ and $\dfrac{\...
Alex Nguyen's user avatar
  • 1,257
2 votes
1 answer
45 views

Continuity proof epsilon delta $\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$

I'm trying to prove the continuity of $\frac{x\sin(xy)}{\sqrt{x^2+y^2}}$ at $(0,0)$. Is the following working correct, and is the way of writing my proof correct? Let $\delta = \sqrt{\epsilon}$. Then, ...
Layla16's user avatar
  • 93
1 vote
1 answer
87 views

Is the function $(x^2+y^2)\sin{(\frac{1}{x^2+y^2})}$ differentiable at the point $(0,0)$?

EDIT: The question specifies that $f(0,0)=0$ in a piecewise. For the first few parts of a question, I have used the definition of partial derivatives to show that $\frac{\partial{f}}{\partial{x}}$ and ...
Peter Chen's user avatar

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