Questions tagged [discontinuous-functions]

Discontinuous functions in $\mathbb R$ are characterized as being "broken" when pictorially represented in graph form. More generally, a function $f:X\to Y$ is discontinuous at $x\in X$ if there exists an open set $V\subset Y$ such that $f(x)\in V$ and $x\in\operatorname{Bd}(f^{-1}(V))$. Use this tag to ask questions about discontinuous functions on the reals or on other topologies.

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29 views

prove non-continuity using open sets (topology)

In topology, continuity is defined as: A function $f:X\rightarrow Y$ is continuous if the inverse image of an open set in $Y$ is an open set in $X$. I have a problem to use it to check the non-...
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Is the following function $f$ continuous? [closed]

Say we have a function $f(x)$ = $\frac{1}{x}\\$ for all x $\in$ $\mathbb{R}$ such that x $\ne$ 0 and $f(0)$ = 2. Do we consider this function f continuous?
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38 views

What is the intuition behind how discontinuous a derivative can be?

I’ve looked at some answers on this site about how discontinuous a derivative can be, and it seems there are some properties that a derivative must satisfy. Darboux’s theorem tells us that if we have ...
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2answers
55 views

How does $(t^3, t^2)$ represent $x^{2/3}$?

I just learnt about this parameterization and somehow I am not being able to wrap my head around it. How on earth is the curve $\left<x, x^{2/3}\right> = \left<t^3, t^2\right>$. I got ...
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Constructing $\sqrt{\sqrt{1+x^2}-1}$ to be smooth (cancelling the square)

$\DeclareMathOperator{\sign}{sign}$ Is there a way to rewrite $f(x)=\sign(x)\sqrt{\sqrt{1+x^2}-1}$ using (smooth) elementary functions? As far as I can see the function seems infinitely ...
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1answer
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How to prove that the n-th root is not continuous using the Fundamental Group

I need to prove that the function $g:\mathbb{C}^∗ \rightarrow \mathbb{C}^*$ such that $(g(z))^n=z$ is not continuous using the fundamental group. I tried to use the argument in the question: How to ...
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1answer
65 views

How to prove that the complex logarithm is not continuous using the Fundamental Group

I need to prove that the function $f:\mathbb{C^*} \rightarrow \mathbb{C} ; \exp{(f(z))} = z$ is not continuous using the fundamental group. I´ve found this Does every continuous map induce a ...
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2answers
54 views

Proving Existence of Discontinuity

I need to prove that $f:[0, 1] \to \Bbb R$ given by $f(x) = \begin{cases} 1, & \text{if $x=\frac{1}{n}$ for any positive integer $n$} \\ 0, & \text{otherwise} \end{cases}$ has an infinite ...
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42 views

Discontinuity multiplied by a distribution. Does the sum of undefined pieces yield a defined one?

Henceforth, let $\delta$ be the Dirac delta and $\theta$ the Heaviside. While the multiplication, in the distributional sense, $\langle\psi\delta,f\rangle$ with smooth $\psi \in C^{\infty}(\mathbb{R}...
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37 views

Examples of continuous/discontinuous functions where one's intuition fails

If I understand correctly, the formal definition of a continuous function $f:X \to Y$ is when $$ \lim_{x \to c}f(x)=f(c) $$ Where $c$ is a constant, and $\{x,c\} \in X$. The layman explanation I was ...
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Differentiability and Continuity in Exponential-Gauss Covariance Functions

Updated: In the Exponential-Gauss covariance function: $$ R(\theta_k;d_{ij}^k)=\exp(-\theta_k|d_{ij}^k|^h) $$ where $0<h \leq 2$, what can we say about differentiability and continuity of this ...
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Is $f(x,y,z) = x+2y-3z$ a continuous function? How do I decide without graphing the function?

I am just starting to learn continuity of functions and trying to find out ways how to figure out if a given function is continuous or not. So far I have found that we will have to graph a function to ...
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114 views

Riemann Integrals vs Area Under Curve intuition

I have the (infinitely) discontinuous function defined by $$f(x)=\begin{cases} \frac{1}{n} & x\in(\frac{1}{n+1},\frac{1}{n}]\\ 0 & x=0 \end{cases}$$ Who's graph is as such: Now, by inspection ...
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A sequence of discontinuous functions that converges uniformly to a continuous function

The definition I have been given on my course for uniform convergence is that A sequence of functions $f_n:D\to\mathbb{R}$ converges uniformly to a function $f:D\to\mathbb{R}$ if the real sequence $$|...
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Removing discontinuity from harmonic mean of non-negative numbers

I am taking the harmonic mean of two numbers which happen to be non-negative. I can write them as $|x|$ and $|y|$. The harmonic mean is, $$ H = \frac{1}{\frac{1}{|x|}+\frac{1}{|y|}} $$ I can remove ...
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Can you help me check the contruction of Operator J'_1: H^1(Omega\Gamma) -> H^1(Omega) right or not? where Gamma is a curve in Omega

We suppose that $\mathcal{H}= L^2(\Omega)$, $\mathcal{\widetilde{H}}= L^2(\Omega\backslash\Gamma) $ and $ \mathcal{H}_1=H^1(\Omega)$, $\mathcal{\widetilde{H}}_1= H^1(\Omega\backslash\Gamma)$. It ...
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proof -divergence series

I know that the series 1/n and -1/2n is divergent but how do i prove it?
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1answer
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Question on discontinuities when denominator has roots but factors out with numerator.

Two questions and the answers that I came across have shaken my fundamental concepts of continuity. They are as follows: $$f(x) = \frac{4-x^2}{4x-x^3}$$ $$h(x) = \frac{(x^2-4)(x-1)}{x^2-a}$$ a. $f(x)...
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1answer
62 views

Does monotonic function imply continuity?

My formula book states that in order to apply variable substitution on an integral, i.e use the formula: $$ \int_a^b f(x)dx = \int_{g^{-1}(a)}^{g^{-1}(b)} f(g(t))g'(t)dt $$ $g(t)$ needs to be a ...
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Time spent above 0 for time-changed càdlàg functions

Let $\psi \in D([0,T],\mathbb R)$, the Skorokhod space of real-valued càdlàg functions (right continuous with left limits) on $[0,T]$. Let $\Gamma$ be the set of time-changes $\gamma$ that are ...
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3answers
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$f(x)=x$, if $x$ is rational and $f(x)= -x $ if $x$ is irrational.

The function is defined on $[a,b]$. I have proved that the function is continuous in $0$ using the definition with $\epsilon$. But I want to proof when the function is not continuous, I think that ...
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Any $F_\sigma$ set in $\mathbb R$ is a set of discontinuity of some function $f$.

Consider the following lemma: Lemma Let $A\subset \mathbb R$ be an $F_\sigma$ set,then $\exists f:\mathbb R\to \mathbb R$ such that $D(f)=A$. Proof: $A=\bigcup\limits_{n\in \mathbb N} A_n$ where $...
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Find the points of continuity and discontinuity. [duplicate]

Let $f:[0,1]\to \mathbb{R}$ be defined by $f(x)=\frac{1}{q},\; \;\; x=\frac{p}{q}\in[0,1], p,q \in \mathbb{Z}^+$ and $f(x)=0,\;\;\; otherwise.$ Find the points of continuity and discontinuity. ...
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Is universal approximation theorem of Artificial neural network (ANN) can be extended for discrete functions?

I know that the universality of ANN is applicable to continuous functions, can it be used for the discrete functions in the following case: Given a discrete function $F$ with domain {$0,1$}$^m$, ...
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1answer
32 views

How to Prove Characteristic Function of the Rationals is Discontinuous Using Sequences

The characteristic function of the rationals $$\chi_{\mathbb{Q}}(x)=\begin{cases}1&x\in\mathbb{Q}\\0 & x\not\in\mathbb{Q}\end{cases}$$ is discontinuous for all $x$. I have seen a proof that $\...
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1answer
25 views

Find the jump of force of mortality under Balducci assumption

I need to find the jump of $\mu_x$ (force of mortality) under Balducci assumption at the point n $\in \Bbb N$. Under the assumption of Baoducci force of mortality function has the form: $$\mu_{x}= \...
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2answers
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Optimization over a conic

Find global maximum and global minimum of $$f(x,y) = \frac{\ln \left(\frac54y^2+(x-2)^2 \right)}{\sqrt[3]y}$$ over $$D = \left\{ (x,y) \in \mathbb{R}^2 :\frac14 y^2 + (x-2)^2 = 1 \right\}$$ My ...
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33 views

When I solve 1st linear ODE with p(t) jump discontinuous, why should I find continuous y?

enter image description here My textbook said Discontinuous Coefficients. Linear differential equations sometimes occur in which one or both of the functions p and g have jump discontinuities. If t0 ...
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5answers
85 views

Why can I not integrate $\frac{1}{x^2}$ over $0$?

The question is: $$\int_{-2}^{1} \frac{1}{x^2} dx$$ My solution is by fundamental rule is : $$\tfrac{1}{-2+1} \cdot x^{-2+1} \Big|_{-2}^1 = \frac{-3}{2}$$): But the solution is said ...
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11 views

Discontinuity of periodic functions with positive and non-summable Fourier coefficients

Let $f : \mathbb{T}^d \rightarrow \mathbb{R}$ be a periodic square integrable function defined over the $d$-dimensional torus $\mathbb{T}^d = [0,2\pi]^d$ with Fourier sequence $(c_n(f))_{n\in \mathbb{...
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1answer
33 views

What is meant by “discontinuous” here?

I am reading signal processing first by Mcclellan In chap 3,last para of article 3.1.2, I came across a term "discontinuous" as shown underlined in attached photo What is meant by it in the context ...
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8 views

Proving an equivalent characterisation of non existence of left and right hand limit for a bounded function

Here's the lemma that I've been trying to prove: Suppose $f: A \to \mathbb{R}$ is bounded and there is a $\delta >0$ such that $(a, a+\delta) \subset A$. Then $\lim_{x\to a^{+}} f(x)$ fails ...
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1answer
31 views

Mean value theorem (integral version) for function with bounded variation

On Wikipedia I find the following version of the integral mean value theorem (MVT) for continuous functions: If $f\colon[a,b]\to\mathbb{R}$ is continuous, then there exists a $c\in(a,b)$ such that $...
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1answer
28 views

If a double integral does not converge absolutely, does that mean the function is discontinuous? [closed]

An assumption of Fubini's theorem is that: $$\displaystyle{\iint_{R} |f(x,y)| \thinspace \mathrm{d} A < \infty}$$ If this assumption is not met, and the integral of the absolute value of the ...
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21 views

Counterexample related to continuity,restriction function.

In Searcoid, I found a problem which asks a question to the reader that if $f:A\cup B\to Y$ is a function such that $f|_A$ and $f|_B$ are continuous and $A\cap B\neq \phi$ where $A,B\subset X$. Then ...
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1answer
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To prove $E_r$ is closed

I came across this theorem which states, Let f:$R^1$->$R^1$. For any r>0 let $E_r$ be the set of all $a \in R^1$ such that, $\omega$[f;a]$\geq$ 1/r. Then $E_r$ is closed. Here I am finding it ...
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1answer
29 views

Oscillation of a function over a bounded open interval

I came across this definition when studying about discontinuous functions on $R^1$ Let f:$R^1$->$R^1$.If J is any bounded open interval in $R^1$, we define $\omega$[f;J] (called the oscillation of ...
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If a function is continuous everywhere and $f(x)=0$ for all rationals then prove that $f(x)=0$ for all reals.

A local book problem: A function $f:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$ and $f(x)=0$ for all $x\in\mathbb{Q}$. Prove that $f(x)=0$ for all $x\in\mathbb{R}$. There was a hint in the ...
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19 views

Homogeneous Markov Chain with Continuous time

Can a transition matrix in a homogeneous Markov chain with continuous time turn out to be discontinuous in t?
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How much integral has been generalized from the time of Riemann?

Is the concept of integral developed so much in higher analysis so that integral of every everywhere discontinuous real function of a real variable defined on the segment exists and is unique? If not ...
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1answer
39 views

Which step was wrong. Please help on this question.

I choose step 5 as wrong in this question, but since I failed the test. I need the right answer Edit: I know x=3 is not possible but still want to know exactly which step is wrong? This is the ...
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3answers
50 views

Graph of $xy=0$ has discontinuity at $(0,0)$ - (undefined, 0).

In finding an explicit expression for y, by diving by $x$, do you implicitly assume that $x$ is not equal to zero because that would give $\frac{0}{0}$? So, there is a removable discontinuity at $(0,0)...
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121 views

Unique increasing solution to a separable differential equation (piecewise $C_1$)

I want to find the increasing function $y(x): [0,1] \rightarrow [0,1]$ which is defined implicitely as the solution to the following equation: $f_1(x) = f_2(y(x)) \quad \forall x \in [0,1]$ ...
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1answer
39 views

absolute asymptotic condition number uniqueness differential equation

Suppose that we have a piecewise, differential equation. Say $\frac{dx}{dt} = \begin{cases} x \sin \frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ $x(0) = 0$ I like to ask if the ...
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27 views

What is it called when a function's maximizer is discontinuous in a parameter value?

Suppose I have function: $f(x;a)$ where $a$ is a parameter. The maximizer $x^{*}(a)=\text{argmax}_xf(x;a)$ is discontinuous in $a$. Is there a general term for this? I know this question is vague, ...
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1answer
54 views

Is this map from $(S^1)^n$ (n-copies of $S^1$) into $S^1$ continuous?

Let $S^1$ be the $1$-dimensional sphere given by $S^1= \{e^{i\theta} \ | \ 0 \leq \theta < 2 \pi \}$. Define a map $f:(S^1)^n \longrightarrow S^1$ given by $$f(e^{i \theta_1}, \ldots ,e^{i \...
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52 views

Convolution with discontinuous function

I want to calculate a convolution of a discontinuous function $f$ with a continuous function $g$. For example $$(f*g)(t) = \left(\dfrac{t+a}{t^2-b^2}\right) * \left(\dfrac{t-c+id}{(t-c)^2+d^2}\right)$...
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1answer
52 views

Is the function $f$ continuous at $(0,0)$??

Let $\mathbb R^2$ endowed with the euclidean norm $||\cdot ||_2$. Let $f : \mathbb R^2 \to \mathbb R$. Assume that there exists two real-valued sequences $\{x_n\}_{n>0} $ and $\{y_n\}_{n>0}...
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65 views

Is the function $f$ continuous at $(0,0)$?

Does $f$ have a limit as $(x,y)$ tend to $(0,0)$? If yes calculate the limit. $$f(x, y) := \dfrac{1−\cos (xy)}{xy^2}$$ [ Hint: consider the variable $t := xy$ ] So first I showed that the function ...
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1answer
30 views

Discuss whether the given function is smooth, piecewise smooth, continuous, piecewise continuous

Discuss whether the given function is smooth, piecewise smooth, continuous, piecewise continuous, or none of these on the interval $\left [ -\pi ,\pi \right ]$ $f(x)=\left \{ \begin{matrix} 1 & ...

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