# 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.

208 questions
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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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### 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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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, $f_{\Phi_{... 1answer 30 views ### How can a discontinuous function belong to$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) $$u(x,y) = \... 2answers 70 views ### Any countable set of real numbers is set of discontinuities of some monotone function. I am studying for a final exam and have come across the following old exam question: Prove that any countable set of real numbers is the set of points of discontinuity of some monotone function. The ... 0answers 12 views ### f:[0,1]\to\Bbb{R} has property- \forall y\in\Bbb{R}, either \nexists x\in[0,1] s.t. f(x)=y or \exists exactly two such points in [0,1]. My whole question looks like- A real valued function f:[0,1]\to\Bbb{R} has the property that \forall y\in\Bbb{R}, either \nexists x\in[0,1] s.t. f(x)=y or \exists exactly two such ... 0answers 16 views ### How to find the curve of discontinuity for a second order linear PDE Question: Solve the following PDE in the quadrant x>0 \; , \; y>0.$$\frac {\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}=-1u=\frac{\partial u}{\partial y}= 0 \qquad \... 2answers 16 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 ... 0answers 51 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 ... 0answers 46 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 ... 2answers 29 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 ... 2answers 61 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 ... 1answer 80 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$0$. ... 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 ... 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 ... 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 ... 0answers 34 views ### 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,$\...
I want to learn about implementing convection-diffusion simulations using discontinuous Galerkin (DG) finite element methods to solve $$\dfrac{\partial c}{\partial t} = \nabla \cdot \mathbf{J},$$ ...