Questions tagged [piecewise-continuity]

For use when asking questions about piecewise continuous functions, their properties, or the functional branch itself.

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11 views

Limit Derivative of solution to linear complementarity problem at discontinuity

Consider the linear complementarity problem with $\mathbf{M}$ a $P$-matrix and a vector $\mathbf{q}$ with unknown $\mathbf{z}$: Find $\mathbf{z}$ such that $\mathbf{M}\mathbf{z}+\mathbf{q} \geq \...
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40 views

Absolute value function definition

The standard definition of is $f(x)=\begin{cases}x,& x\geq 0\\-x,&x<0\end{cases}$. I am wondering what the problem is with the definition $f(x)=\begin{cases}x,& x,\geq 0\\-x,&x\leq0....
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Derivative of piecewise function at a point (function is given)

$f(x)= \begin{cases} x^2 & \text{if } -2 ≤ x ≤ 2, \\ 2x & \text{if } x > 2. \end{cases}$ I am finding the derivative at $x = 2$. I am saying that the derivative is $4$ because $x =2$ is on $...
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28 views

Continuity of piecewise function involving rationals/irrationals and Cantor set

I'm struggling to determine the continuity of the following function: $$ f(x)=\begin{cases}0 \quad \text{if $x \in \mathbb{Q}\cap D,$}\\ x^3 \quad \text{if $x \notin \mathbb{Q}\cap D$;} \end{cases}$$ ...
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Convolution, How to develop this piecewise?

This is an exercise from my book on hilbert spaces. I don't understand the last step. Can someone develop the procedure to get that piecewise?. Convolution $\left(f\ast g\right)$ Indicator function $\...
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Nondecreasing and piecewise-continuous => piecewise-continuously-differentiable?

I'm interested to know if a piecewise-continuous monotone function from $\mathbb{R}$ to $\mathbb{R}$ is also piecewise-continuously differentiable. I mean piecewise in the sense that there is a finite ...
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21 views

About piecewise function without the heaviside step function .

Can a piecewise function of a single variable without indefinite limit can be always concentrate into a single variable function (without the heaviside step function)? Well I take an example : Let $0....
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38 views

Differentiability of ReLU function

Given a ReLU function $f(x) = \max(0,x)$, one can show it is not differentiable at $x = 0$ on domain $\mathbb{R}$. If we restrict $x \in [0, \infty)$, the function would become $f(x) = x$. Is it also ...
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Convert piecewise function with ranges to a single mathematical formula

For a programming task (an SQL expression) I need a function made of mathematical operators ($+$, $-$, $*$, $floor$, $ceil$, and comparison operators returning value $0$ (false) or $1$ (true)...) that ...
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40 views

How to find specific functional square roots (like half iteration of exp, ln, ...)?

Curious about intermediate mean between arithmetic and geometric, I noticed that all follow pattern $M(\space f \space,\space x[1,2,...,n] \space,\space w[1,2,...,n] \space)=f^{-1}(\frac{w[1]\space*\...
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91 views

Identify where $ f(x)= \sqrt{\frac{1-x}{|x|}}$ is continuous

So I am trying to identify where $\sqrt{\frac{1-x}{|x|}}$ and I want some assistance if my my reasoning is right. $$f(x) = \begin{cases} \sqrt{\frac{1-x}{x}} ,& x \geq 0 \\ \sqrt{\...
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23 views

Geometry of orbits in regular grazing point.

Consider a piecewise smooth impact dynamic system of a field $F \in C ^ r $, with discontinuity boundary $ \Sigma = H ^ {-1} (0) $, where $0$ regular value of H. Let $ x ^ * $ be a regular grazing ...
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Lie Derivates in piecewise smooth impact dynamic system

Consider a piecewise smooth impact dynamic system of a field $F \in C ^ r $, with discontinuity boundary $ \Sigma = H ^ {-1} (0) $, where $ 0 $ regular value of H. Let $ x ^ * $ be a regular grazing ...
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40 views

Is this conclusion on stone-weierstrass approximation true?

Prove or disprove that it's possible to approximate the functions in $C([a,b],\mathbb R) $ uniformly with arbitrary error $\epsilon$ by functions in the algebra $\mathcal A$ generated on $[a, b]$ by ...
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44 views

Induced continuity or differentiablity on a line on a square

Let $k\in\mathbb N_0$, $A:=(-1,1)$ and $f:A^2 \to \mathbb R$ be a function with the following properties: For every fixed fixed $y \in A$ the function $f(\cdot,y)$ is $C^\infty$. For every fixed ...
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27 views

Construct a sequence of $C^1$ functions that converges to a piecewise linear function

Consider the piecewise linear function defined by $$y_0(x)=\begin{cases}0&x\in[-1,0]\\x&x\in(0,1]\end{cases}$$ I want to find a sequence of function in the domain $$D=\{y\in C^1[-1,1]\big|y(-1)...
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35 views

Can i represent derivative as Lipschitz constant, if a function defined on interval (a,b]?

I am working with some piecewise linear functions and its' intervals defined as $I_j$=($x_j$,$x_{j+1}$]. I want to use the derivatives of sub-functions as Lipschitz constants, but in order to do that, ...
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Proving the integration of piecewise cubic functions as 0

Given that $g(x)$ is a piecewise cubic function, and $h(x) = \bar g(x)-g(x)$. I am struggling to show that $\sum_{i=1}^{N}\int_{x_{n-1}}^{x_n}g'''(x)h'(x)dx = 0$ I understand that $g'''(x)$ will be a ...
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Showing Differentiability/Continuity at endpoints of closed interval?

I am given the function $\gamma:[-1,\frac{\pi}{2}] \rightarrow \mathbb{C}$ $\gamma(t) = \begin{cases} t+1 & \text{for $-1 \leq t \leq0$} \\ e^{it} & \text{for $0 \leq t \leq\frac{\pi}{2}$} ...
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What is the meaning of this notation for a function?

A contour or path is a continuous mapping $\gamma:[a,b]\rightarrow \mathbb{C}$ which is piecewise continuously differentiable, i.e., there exist $a=a_0<a_1<...<a_n=b$ such that $\gamma_{|[a_{...
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Considerations when creating piecewise functions

I am asking about the dos and don'ts of piecewise expressions. First of all, there's the question of completeness versus clarity. Being as complete as possible can lead to long piecewise expressions ...
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74 views

Using 3D Piecewise Functions to Model a Rollercoaster

I am designing a roller coaster using functions (ie. linear, cubic, logarithmic, trigonometric). At some point, one of the parts of the rollercoaster does not follow a two dimensional graph, but ...
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Defining a Minimum cost-flow problem with piecewise costs

I need to formulate an Integer Linear Programming model for a Minimum cost-flow problem over a graph without constraints on edges. The cost over edges isn't linear but piecewise, given by: $c(x_{ij})=\...
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29 views

Defining a time varying uniform charge density

I am having trouble identifying the best manner to define my problem. More specifically, suppose one has a three dimensional, spherical surface charge density. The density is uniform, meaning it is ...
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3answers
72 views

How to write a formula for a piecewise function that contains a single separate point using Heaviside function?

Here and here I saw how to rewrite a piecewise function using the Heaviside function. Thus, if I a have a function that looks like this: I can write it down as: $$y(t) = 1 \cdot [H(t) - H(t-1)] + 2 \...
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functions minimized at extreme value.

Given two ordered sequence $a_1, a_2, \cdots, a_n$ where $a_i \leq a_{i+1}$, and $b_1, b_2, \cdots b_n$ where $b_i \geq b_{i+1}$ and $a_i, b_i \in \mathbb{R}$. I can construct a new sequence $S$ by ...
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Calculating the partial derivatives of wierd function

Define $ f:\mathbb R^2\to\mathbb R $ by $ f(x,y) := \begin{cases} \frac{2 x^2 -4 x + 6 y^3 - 13 y^2 + 8 y +1}{x- y}, \quad &\text{if }x\neq y; \\ 0,\quad &\text{if }x=y. \end{cases} $ I'm ...
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Does order of differentiation matter for piece-wise smooth functions?

$\frac{\partial^2 f}{\partial x \partial y}= \frac{\partial^2 f}{\partial y \partial x} \forall f \in C^\infty $ Is the same true for functions that are piece-wise smooth?
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68 views

Why this piecewise function doesn't have a removable discontinuity?

In the third function from this image. The function when $x$ is not equal to $2$ can be transformed into $ x+1 $, resulting in having a value of $3$ at the point $ x = 2 $. Why is this not considered ...
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Prove a piecewise function is differentiable at $ x = 1 $ [closed]

Use the difference quotient definition of the derivative, $$ f ' ( a ) = \lim _ { x \to a } \frac { f ( x ) - f ( a ) } { x - a } \text , $$ to show that $$ f ( x ) = \begin {cases} x \ln x \text , &...
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37 views

Is this piecewise defined function in two variables lipschitz continuous?

According to Picard–Lindelöf theorem, IVP $$\begin{cases}y^\prime(t) = f(t,y(t))\\y(x_0)=y_0 \end{cases}$$ has a unique solution if $f$ is lipschitz continuous. What if my ODE contain piecewise ...
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69 views

How could one approximate a flat piecewise function using trigonometric identities?

I have a flat piecewise function that is a on the interval $-L<x<L$, and 0 outside of this boundary. Perhaps $a=1/(2L)$, or perhaps not. However, I am looking ...
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33 views

Convergence of the Fourier integral at a point

I'm studying this theorem on Zorich II: (Convergence of the Fourier integral at a point) Let $f : \mathbb{R} → \mathbb{C}$ be an absolutely integrable function that is piecewise continuous on each ...
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Should the approximated values automatically be increasing when the experimental data is increasing?

I am trying to approximate groups of discrete data which are monotonically increasing by continuous peicewise linear function. The objective is minimizing the sum of absolute error. Numerically, I ...
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50 views

Multidimensional piecewise functions with unknown breaks

I am trying to identify unknown breaks in multidimensional continuous piecewise functions. I have a dataset with values p for each point(x,y,z) that I am attempting to fit a continuous piecewise ...
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29 views

Calculating distance function from a piecewise velocity function (pre-calculus)

This is my very first question here. I am self-studying maths and I am currently doing pre-calculus from Gilbert Strang's textbook "Caculus 1". So I apologize in advance if this question ...
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How do i prove that this LP is equivalent to a piecewise linear function?

Let be $$z(e)=min \sum_{k=1}^Rp_k\sum_{j=1}^M q_j(\sum_{i=1}^M c_ig_{ijk}+c_\delta\delta_{jk})\\ s.a \sum_{i=1}^M g_{ijk}+h_{jk}+\delta_{jk}=d_j\\ 0\leq g_{ijk}\leq\overline{g}_{ik}\\0\leq h_{jk}\leq\...
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153 views

Wolfram alpha plot of a piecewise function with 2 domains. [closed]

So, Wolfram Alpha is capable of plotting two functions in one figure. But I am trying to express a piecewise function that has a different form on two (touching) domains. In C code: ...
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Existence and uniqueness of piecewise differential equations

There is a comprehensive theory concerning the existence and uniqueness of the solution of differential equations $$ \frac{d}{dt} x(t) = F(t, x(t)),$$ with initial condition $x(t_0)=x_0$, where $F(t,x)...
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is this function continuous at $x=1$ and $x=-1$ , and why?

Let $f\colon \mathbb{R} \to \mathbb{R}$ be the function $$ f(x) = \begin{cases} \ x \exp\left(\frac{2x}{x^2-1}\right) &\text{if }x \in \mathbb{R} \setminus\{-1,1\},\\ \ 0 &\text{if }x = 1 \...
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Positivity of the solution of a differential equation

Let $a(x)>0$ be a piecewise continuous function defined on $[0, \infty)$ and assume for a differentiable function $f: [0, \infty) \rightarrow R$ we have, $$ \frac{d}{dx}f = a(x) + a_0\,f+ a_{1}\,f^{...
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Can every piecewise linear function be exactly realized as a neural network?

Can every continuous piecewise linear function $[-1,1]^k \rightarrow \mathbb{R}^n$ be written as a composition of the following building blocks: Affine map: $x \mapsto Ax + b$ for some matrix $A$ and ...
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1answer
36 views

Double derivative of piecewise continuous function

The following function is defined for $x \in (-\infty,\infty)$ \begin{equation} f(x) = \begin{cases} x & x<1 \\ 1 & x \geq 1 \end{cases} \end{equation}0 The function is obviously continuous....
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Finding a Nash Equilibrium of a Two-Players Game based on Best Response Functions - Taking a partial derivative from a piecewise, weird, function!

I need to find a nash equilibrium of a two-player game based on their best response function. The problem I have is that the functions I'm dealing with are somehow weird and hard to work with! So, let ...
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Solving a Chain of Convolutions Yielding Piecewise Functions

I want to calculate the convolution of a function $f(t)$ with two other functions $M_1(t)$ and $M_2(t)$, i.e. $f(t)*M_1(t)*M_2(t)$. The functions are as defined below where $u(t)$ is the step function....
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31 views

Gradient vectors for piecewise smooth function

Suppose I have a non-empty open set $A\in\mathbb{R}^N$ with a piecewise smooth boundary $\partial A$. For any point $p\in \partial A$ at which $\partial A$ is locally smooth, let $\vec{N_p}$ be the ...
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37 views

Piecewise smooth set does not contain interior points

In the middle of page 4 of this paper the author states : It is obvious that any set of dimension less than n which is piecewise-smooth in V does not contain interior points. which I do not ...
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80 views

The regularity of a piecewise constant function in Sobolev spaces

I want to know what the "highest" regularity is for a piecewise constant function. For example: $$ f(x)=\left\{\begin{align*} &1, & x\in [0,1),\\ &0, & x\in (-1,0).\end{align*...
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32 views

Piecewise function calculation with variable in integration interval

I have this: $(f*g)(x)=\int_{x-\frac12}^{x+\frac12} f(t)dt$ where $t = x-y$ and $f(x)$=\begin{cases} 0&\text{if}\, x\ < a\\ \exp(-x)&\text{if}\, a \leq x \ \ \end{cases} I have ...
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57 views

Continuity in the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is as follows: $\frac{d}{dx}\int_{a}^{x} f(t)dt = f(x)$ I know that to integrate a function $f(t)$, it must be piece-wise continuous. However, I'm not too sure ...

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