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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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How can I find the value of this [pathological] function?

A few months ago, while attempting to create a parameterization of the Hilbert curve, I discovered an interesting function, given by the summation... $$f(x)=\sum_{n=1}^\infty \frac{\text{sgn}\left(\...
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2answers
71 views

Is composition of surjective continuous function with discontinuous function discontinuous?

Let $I_1,I_2,I_3$ be intervals $\subset \mathbb{R}$. Suppose $f:I_1 \to I_2$ is a surjective continuous function and $g: I_2 \to I_3$ is a discontinuous function. Must the composition $g \circ f$ be ...
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1answer
16 views

How to determine whether a piecewise function with conditions instead of equations has removable discontinuities?

I have to explain whether the piece-wise function below has any removable discontinuities. I am confused because, as far as I know, to determine whether there is a removable discontinuity, you need to ...
2
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1answer
42 views

Jump discontinuity in antiderivative

I am dealing with integral $I=\int_0^{\pi}dx\frac{\sin^2x}{a^2+\sin^2x}.$ I know the answer is $I=\pi(1-\frac{a}{\sqrt{1+a^2}}).$ However I am having some uncertainty getting there. I know that ...
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1answer
57 views

How to solve $f(-4)$ if $f(x)=\frac{x^3+64}{x+4}$?

I need to solve $f(-4)$ if $f(x)=\frac{x^3+64}{x+4}$. I have changed the form of the function, but for some reason I'm still not getting the right answer. My Steps: $$\frac{x^3+64}{x+4}$$ apply $(a+...
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2answers
50 views

Find $f(7)$ if $f(x)=\frac{x-7}{|x-7|}$?

I am trying to find $f(7)$ if $f(x)=\frac{x-7}{|x-7|}$. The problem I'm having, is that I don't know how to rewrite a function with an absolute value so that $f(7)$ exists. I have tried multiplying ...
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4answers
34 views

Remove the discontinuity of $f(x) = \frac{x^2-11x+28}{x-7}$ at $f(7)$?

I need to remove the discontinuity of $f(x) = \frac{x^2-11x+28}{x-7}$ at $f(7)$, and find out what $f(7)$ equals. I am not sure what I've done wrong, but I'm getting 33, which the website I'm using ...
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1answer
55 views

Counterexample for a continuous function

Continuity definition for expectation: For all $\epsilon>0$, there exists a $\delta>0$ such that the following holds $$\forall x~|F(x)-G(x)|\leq\delta \implies |E_F[X]-E_G[X]| \leq \epsilon. $$ ...
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0answers
22 views

Optimization of a bounded discontinuous function

Suppose an unknown function of real variables, $f:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times ...\times \mathbb{R}\rightarrow\mathbb{R}$, where it takes multiple inputs $x_1,x_2,...,x_{35}$ and ...
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1answer
80 views

Show that f is continuous at all c ∈ R \ Z and discontinuous at all c ∈ Z.

For any x ∈ R define the floor of x, denoted [x],to be the largest integer y with y ≤ x. Then define a function f : R → R by f(x) =[x]. Show that f is continuous at all c ∈ R \ Z and discontinuous at ...
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1answer
22 views

Compact subset of the image of a continuous function

Given complete metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, a continuous function $f:X \to Y$, and a compact subset $K \subset f(X)$, I would like to know if it is possible to claim that there exists a ...
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1answer
26 views

Proving that a function is discontinuous using sequential definition

I am struggling to understand how to prove that a function is discontinuous using the sequential definition. Here is a particular example from my textbook where some clarification might help. Let f(x)...
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1answer
49 views

Can a derivative of a real valued function have uncountable points of discontinuity?

Suppose $f$ be a real-valued function, such that $f'$ exists everywhere in the domain. I am thinking about the problem in following steps- 1) Can $f'$ have jump discontinuity?-No, since if it has jump ...
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7answers
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How can a function with a hole (removable discontinuity) equal a function with no hole?

I've done some research, and I'm hoping someone can check me. My question was this: Assume I have the function $f(x) = \frac{(x-3)(x+2)}{(x-3)}$, so it has removable discontinuity at $x = 3$. We ...
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1answer
24 views

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
34 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 ...
2
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1answer
67 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
17 views

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
34 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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0answers
49 views

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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0answers
63 views

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
29 views

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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0answers
13 views

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
36 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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0answers
37 views

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):=...
3
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0answers
45 views

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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0answers
48 views

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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0answers
25 views

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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64 views

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
25 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
445 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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0answers
47 views

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
34 views

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
31 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
28 views

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
147 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
50 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
38 views

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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2answers
92 views

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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0answers
19 views

$f:[0,1]\to\Bbb{R}$ 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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0answers
20 views

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
17 views

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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0answers
104 views

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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0answers
47 views

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 ...
0
votes
2answers
36 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 ...
2
votes
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 ...
2
votes
1answer
106 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 $0$. ...
0
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1answer
52 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 ...