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Questions tagged [discontinuous-functions]

For questions about discontinuous functions, a function which for certain values or between certain values of the variable does not vary continuously as the variable increases.

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On the physical sample of “ discontinuous differential equations ”

The following source contains several physical examples: https://arxiv.org/pdf/0901.3583.pdf. How can I get the differential equation in "Example 2: Brick on a frictional ramp"? How can I grasp this ...
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Is exceptional set countable or uncountable?

In Pollicott and Simon paper https://www.jstor.org/stable/2154881?seq=4#metadata_info_tab_contents Corollaty 1 gives cardinality of set of discontinuity points (exceptional set) is $ \aleph $. And in ...
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Show that any bijection f:[0,1] to (0,1] has infinitely many discontuinty? [duplicate]

I am struggling with this question. I could not even find such a function! Could you give an example. Thank you for your help and advice.
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Is sliding mode control complete in mathematics?

Consider a double integrator control system $$\begin{cases}\dot{x}_1 = x_2,\\\dot{x}_2=u,\end{cases}$$ where $\dot{x} := \mathrm{d}x/\mathrm{d}t$. Then we apply the sliding mode control $u = -\beta(...
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Points of discontinuity and non differentiability of $| \sin(\pi/x)|$?

What are the points of discontinuity and non-differentiability of $| \sin(\pi/x)|$? I tried finding out the points of discontinuity for the function but couldn't understand why would a mod function ...
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How to solve $\dot{x}=f(x)/||f(x)||$?

How to solve the following ODE? \begin{equation} \frac{d}{dt}x=\frac{f(x)}{\|f(x)\|}, \end{equation} where $x: \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f: ...
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Discretization of continuous model with white noise to use Kalman filter later

I have this system which describes dynamics of a car in 2D space. The dynamics are governed by Newton's law g(t) = ma(t). The final task is to use Kalman filter on discretized system to estimate it's ...
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The oscillation of a bounded function at a point

Enumerate the rationals in $[0,1]$ (ie. $\mathbb{Q}\cap[0,1]$) by $q_n$. Define $f:[0,1]\to\mathbb{R}$ by $$ f(x)= \begin{cases} 1/n & \text{if } x=q_n \text{ for some }n\\ 0 & \text{...
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1answer
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If $P(X = c) > 0$ show that $F_{X}$ is discontinuous at $c$

Let $(\Omega, \mathcal{F}, \mathbb P)$ and $c \in \mathbb R$ so that $\mathbb P(X = c) > 0$ Show that the distribution function $F_{X}$ of $X$ has a point of discontinuity at $c$. My ideas: ...
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1answer
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Functions that Tend To Non-Smooth Functions as Some Parameter Tends to Infinity

I recently saw a post in which the query was about a function that tends to the Dirac delta function as a parameter in it tends to infinity. The function chosen was $${(1+\cos x)^n\over C}$$ as $n\to\...
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1answer
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Show that $f$ is not Riemann integrable on $[0,1]$

If $x$ is any rational number, $f(x)=0$. If $x$ is any irrational number, $f(x)=1$. I know that $f(x)$ oscillate between $0$ and $1$ on $[0.1]$. But I have not idea why it isn't integrable on $[0.1]$...
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Example showing $\lim\limits_{x \to x_0} xf(x) \neq x_0\lim\limits_{x \to x_0} f(x)$

I can looking for a simple example to illustrate $\lim\limits_{x \to x_0} xf(x) \neq x_0 \lim\limits_{x \to x_0} f(x)$ For example I have tried $f(x) = x-1, x_0 = 1$ hoping that I would get a zero on ...
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distribution associated with a discontinuous function

Let $f\colon\mathbb{R}\to\mathbb{R}$ be such that, for every $n\in\mathbb{Z}$, $f$ is differentiable on $\left(n,n+1\right)$ and $n$ is a discontinuity of first kind of $f$. We define $$T_f(\phi)=\...
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Multivariable calculus discontinuities question

Question: If D is the set of discontinuities of $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, show that the set of discontinuities of $f_{\Phi_{A}}$ is contained in $D \bigcup \partial A$. So, $f_{\Phi_{...
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How can a discontinuous function belong to $C_B^1(\Omega)$, the space of continuous functions $u$ with bounded derivatives?

Let $\Omega = \{(x,y) \in \mathbb{R}^2 \ : \ 0 < |x| < 1, \ 0 < y < 1\}$ and consider the function $u$ defined on $\Omega$ by (Sobolev Spaces by Adams, page 68, Example 3.10) $$ u(x,y) = \...
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Any countable set of real numbers is set of discontinuities of some monotone function.

I am studying for a final exam and have come across the following old exam question: Prove that any countable set of real numbers is the set of points of discontinuity of some monotone function. The ...
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$f:[0,1]\to\Bbb{R}$ has property- $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such points in $[0,1]$.

My whole question looks like- A real valued function $f:[0,1]\to\Bbb{R}$ has the property that $\forall y\in\Bbb{R}$, either $\nexists x\in[0,1]$ s.t. $f(x)=y$ or $\exists$ exactly two such ...
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How to find the curve of discontinuity for a second order linear PDE

Question: Solve the following PDE in the quadrant $x>0 \; , \; y>0$. $$\frac {\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}=-1$$ $$u=\frac{\partial u}{\partial y}= 0 \qquad \...
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Example of a discontinous pseudo contraction mapping

$T: X\to X$ is a mapping with a fixed point $x^*$ with a property $\|T(x)-x^*\|\le \alpha\|x-x^*\|\forall x\in X,\alpha\in[0,1)$, could anyone give an example of such a map but discontinous? or ...
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If $f$ is monotonic on $(a,b)$, the set of points of (a,b) at which $f$ is discontinuous is at most countable.

Now as an undergraduate student, I am studyign baby Rudin. I know the proof of this theorem are already well explained on match stack exchange here, but I have some question about the proof. In page ...
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showing that a continuous function attains each of its values(or each real number in general) exactly 3 times.

The question and its answer is given below: But I am wondering how I can prove that the function $f$ attains each real number exactly 3 times, could anyone show me how can I do this? I ...
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Why the function has removable discontinuity

$f(x) = \begin{cases} 2x-1 & \text{when }x<2 \\ 5 & \text{when }x=2 \\ \frac{1}{2}x + 2 & \text{when }x>2 \end{cases}$ I am learning Calculus But I can't seem to understand why this ...
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2answers
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Real valued function which is continuous only on transcendental numbers

First of all, I am sorry for asking this question. We know that $R$ is uncountable. And also the set of all transcendental numbers is uncountable. How can I construct a function $f(x)$ on $R$ which ...
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1answer
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Is every measure $0$ set a set of discontinuities of a Riemann integrable function?

Let $f:[a,b]\rightarrow\mathbb{R}$ be bounded, and let $D$ be its set of discontinuities. Then Lebesgue's criterion states that $f$ is Riemann-integrable if and only if $D$ has Lebesgue measure $0$. ...
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1answer
27 views

Showing a function is discontinuous using open sets

I am experimenting with the following theorem: A function $f:A\rightarrow B$ is continuous iff $f^{-1}(O)$ is open in A for every open set $O\subset B$. I am trying to find an open set in the ...
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Within a limit, find the constant c.

Find the constant $c$ such that the limit of the following, exists. $$\lim_{x\to 3} \frac{x^2+x+c}{x^2-5x+6}$$ What I've tried So typically to find a limit, you substitute the number that x ...
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How to remove discontinuity from the equation?

I have a function $f$, which have two cases: $$ f(x) = \begin{cases} \frac{ \sin(a + bx) - \sin(a) }{b} &\mbox{if } b \neq 0\\ x \cos(a) &\mbox{if } b = 0 \end{cases} $$ The first part has ...
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Derivative in removable discontinuity with change of variables

I am trying to rigorously prove a result about derivatives when removable discontinuities are involved. Specifically, I have a bi-variate function $f(a,b)$ that is undefined for $a=0$. However, $\...
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Flux Limiter for 2D Discontinuous Galerkin FEM

I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve $$ \dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J}, $$ ...
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1answer
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Is the derivative of a Lipschitz continuous gradient function is a continuous vector function?

Let $f(x)$ be a Lipschitz continuous gradient function, that is $$ \|f'(x)-f'(y)\| \leq \alpha \|x-y\| $$ where $\|\cdot\| $ is Euclidean norm and $x,y \in \mathbb{R}^n$ and $\forall x,y \in \textbf{...
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1answer
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Sum of a infinite number of continuous functions on a set may not be continuous.

I was asked to give an example of series of continuous functions whose limit is discontinuous . I gave the following example: $f_n(x) = x^n - x^{(n-1)}$ . I thought any sequence of continuous ...
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Analytic integration of discontinuous function

I'm trying to calculate the convolution between a Gaussian and a discontinuous function: $$(f*g)(t)=\int_{-\infty}^\infty f(\tau)\,g(t-\tau)\,\mathrm{d}\tau$$ Where $$f(t\geq0)=\mathrm{exp}(-kt),\,f(...
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I am not being able to solve this problem of Continuity. [closed]

Prove that $[x] \sin^2(\pi x)$ is continuous at every integer point and $[x] \cos^2 (\pi x)$ is discontinuous at every integer point.
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Discontinuity - Unsure If Piecewise Equation(s) Have Them

I have a question on whether the functions following have a discontinuity, and if not, what are the points where two functions meet. First, the piece wise equation : \begin{align*} f(x)= \begin{...
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Mapping disconnected sets onto connected sets

I was just thinking a little about functions that map disconnected sets in $\mathbb R^n$ onto connected sets in $\mathbb R^n$. If a function does something like that it seems to me that such a ...
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1answer
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Continuous function of class $C^2$ that consists of cases $0$ if $x\geq 0$, positive if $x<0$

question I am trying to understand the concept of continuity better and for that I wonder, if the following function $$ f(x) = \begin{cases} 0 &\text{if } x \geq 0\\ x^2 &\text{if } x < 0 ...
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On the choice of general Lyapunov functions for discontinuous control

Let us consider the following dynamical system: $$ \dot{X} = A\cdot X + B\cdot u$$ where $X,B \in \mathbb{R}^{n\times 1}$. The considered system is linear, but I think that $A\cdot X$ can be ...
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2answers
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Types of undefined for removable discontinuities and vertical asymptotes

Consider this rational function: $$ f(x) = \frac{x^2 - 2x - 24}{x^2 + 10x + 24} $$ I have been taught that to solve for a removable discontinuity, I find the $x$ values such that both the numerator ...
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Can removable discontinuities be ignored while integrating?

A function needs to be continuous in order to be integrable. However, can single point removable discontinuities be ignored while integrating? I ask this because obviously a point isn't going to ...
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Does piecewise continuity of $f'$ implies that the discontinuities of $f$ are jump discontinuities?

In this pdf, the following theorem is stated without proof: I'm not sure how this mathematically is accurate. My question is: Do the limits $f(x^+)$ and $f(x^-)$ exist, in the last statement? i.e. ...
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Prove that, the map $x\mapsto=[x]\cos^2{\pi x}$ is discontinuous at every integer points. [closed]

I think that the map $x\mapsto=[x]\cos^2{\pi x}$ may have jump discontinuities at every integer point. However I cannot establish that fact in a rigorous manner. Can anybody assist me to find the ...
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Sequence of continuous functions with discontinuous limits

A sequence of smooth functions such as $h_n (x) = arctan(nx)$ has discontinuous limit that depends on x. When $n \rightarrow \infty$, $h_n (x)$ converges to $$\left\{\begin{array}{ccc} -\pi/2, & ...
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2answers
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How to solve for x:$ \left \lfloor{x} \right \rfloor\ - n \cdot \left \lfloor{\frac{x}{n}} \right \rfloor\ = y$

Getting solutions to questions with floor functions, like: solve for x:$$ \left \lfloor{x} \right \rfloor\ - n \cdot \left \lfloor{\frac{x}{n}} \right \rfloor\ = y$$ $$st.: \ x ∈\{0, \mathbb{R^{+}} \}...
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1answer
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anti-derivative not differentiable at any point

Reading about primitives and anti-derivatives, I noticed that primitive functions of non-continuous functions are not differentiable at some point, but the set of non-differentiability is often ...
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1answer
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Find a discontinuous function defined on $\mathbb{Q}$ embedded in $\mathbb{R}$

I am reading Chapter 1 Example 11 of 'Counterexamples in Analysis' by Gelbaum and Olmstead. This section illustrates counterexamples of functions defined on $\mathbb{Q}$ embedded in $\mathbb{R}$ of ...
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A question about the set of discontinuity of a function related to a series representation for $\zeta(3)$ [closed]

In this post I was inspired in a series used by Apéry, see the MathWorld's article Apéry's Constant to define the following function for $0\leq x\leq 1$ $$f(x)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^3\...
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1answer
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Proof that $f(x,y) = (x^2 + y^2)\sin(x^2 + y^2)^{-1/2}$ is differentiable?

Consider the function $$f(x,y) =(x^2 + y^2)\sin\left(\frac{1}{\sqrt{x^2 + y^2}}\right)$$ The partial derivative with respect to $x$ are equal to $$\frac{\partial}{\partial x}f(x,y) =\left\{\begin{...
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1answer
60 views

How is continuity of a constant function defined?

Definition of continuity is that for small changes in the input there should be small changes in the output.Otherwise the function is discontinuous. So,with this definition we can say that a constant ...
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Question on Dirichlet function graph.

enter link description herehttp://mathworld.wolfram.com/DirichletFunction.html Why has Wolfram mathworld shown Dirichlet function like this? Are they just representing the point by making straight ...
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Delta Epsilon proof of discontinuity example

The function is: $\ f(x) = \begin{cases} \frac{7}{100}x^2-\frac{3}{5}, & x\le 1 \\\\ \frac{100x^2-137x+37}{100(x-1)}, & x > 1 \end{cases}$ For $x_0=1$ find an $\varepsilon$>0 such that for ...