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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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What's the difference between a singular function and a singular continuous function?

I am a physicist so I am trying to make sense of definitions. As far I know, a singular function on $[a,b]$ is defined as: $f$ is continuous on $[a, b]$. the derivative $f′(x)$ exists and is zero ...
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Prove discontinuity at given point

Given the function $y= \lim_{n \to \infty} \frac{1}{1+x^n}$ for $x \geq 0$, show that the function is discontinuous at $x=1$? I tried the question , it comes out to be continuous using left hand ...
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1answer
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A function with discontinuities.

Let $y=f(x)$ be a function which is discontinuous for exactly $3$ values of $x$ but defined $\forall x~{\in}~\mathbb{R}$. Let $y=g(x)$ is another differentiable function such that $y=f(x)g(x)$ is ...
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1answer
28 views

Laplace Transform: Piecewise Function Integrability and Existence of Laplace Transform

I am trying to decide whether the function $$f(t) = \begin{cases} 1, & \text{$t$ is even} \\ 0, & \text{$t$ is odd} \end{cases}$$ has a Laplace transform, or is even integrable in the ...
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1answer
52 views

The set of points where a function $f:[a,b]\to\mathbb R$ is discontinuous is Lebesgue measurable

Prove that the set of points where a function $f:[a,b]\to\mathbb R$ is discontinuous is Lebesgue measurable. Lebesgue measure of set $A$ means that for any set $S\in\mathbb R,$ $m^*(S)=m^*(A\cap S)+...
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1answer
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Continuous and Discontinuous Functions - Variation of Dirichlet function

Be $f: R \longrightarrow R $ a real function where: $ f(x) = \begin{cases} x + 1, & \text{if $x \in Q $} \\[2ex] 2, & \text{if $x \in R-Q$} \end{cases} $ Where the graph "jumps" among the ...
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1answer
26 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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Integrability of function with a bounded set of discontinuities of measure zero

Let $f:[a, b]\to\mathbb{R}$ be a bounded function. Suppose that the set $c$ of discontinuity points of $f$ is a closed subset of measure zero. Show that $f$ is Riemann integrable. What I Did: ...
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Multivariable calculus - Continuity

The following function is not defined for $x=0$ and $y=0:$ $$f(x,y) = \frac {xy} {x^2+y^2}$$ Is it possible to add there a function value in such way that the modified function is continuous at zero ...
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1answer
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The set of discontinuities of an increasing right continuous function is closed.

I'm trying to solve a classical problem on right continuous functions: Every right continuous increasing function $ F$ is the sum of a continuous function $C$ and a jump function $J$ (pice-wise ...
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Function with a derivative with removable singularities, minimization problem

Consider $f(x) = \mid x - a \mid^3$ with $x \in [0,b], a \in (0,b)$. We all agree that the function is convex and it is minimized at $x=a$. $f(x)$ is simpler than the functions I actually have, whose ...
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1answer
31 views

Properties of discontinuity of the second kind

Using Rudin's definition of a discontinuity of the second kind for a function. f has a discontinuity of the second kind if either $f(x^+)$ or $f(x^-)$ does not exist. Supposing that $f$ has a ...
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Riemann Integral of discontinuous function

So I know you can integrate some discontinuous functions when the function is discontinuous at a finite number of points. So you can integrate for example this function on a interval [-1,3]: $$ f(x):=...
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Do separately semi-continuous functions have a dense set of semi-continuities?

The connection between separate continuity and joined continuity has been studied quite a lot. In particular, one has (as a special case of a far more general Theorem from here) the following: If $...
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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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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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1answer
21 views

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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1answer
435 views

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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1answer
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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
27 views

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
96 views

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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4answers
42 views

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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2answers
48 views

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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1answer
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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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1answer
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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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2answers
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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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2answers
30 views

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
62 views

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
89 views

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
36 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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4answers
35 views

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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4answers
52 views

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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20 views

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
28 views

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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1answer
38 views

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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1answer
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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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51 views

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
23 views

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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76 views

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 ...