Questions tagged [step-function]

A step function, also known as a simple function, is a finite sum of characteristic functions of bounded intervals. They are often used in measure theory to approximate integrable functions.

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Examples of counterintuitive regulated functions (solution verification)

Questions I am looking for some counterexamples for properties that of functions that are commonly mistaken for being true in cases where they do not hold in general. The specific cases I am thinking ...
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FTC for Step Functions proof

I'm hoping someone could look over my proof attempt of the following claim. The Statement For the step function $\phi$ on the compatible partition $P=\{p_0,...p_k\}$. Then we say that the function $...
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Integration and Measure problem from Shilov and Gurevich Book.(zero measure definition using step functions))

The problem reads: Let F be the closed interval obtained by removing a countable collection of disjoint open intervals from a closed interval [a,b], where the sum of the lengths of such intervals is b-...
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Upper bound on the integral of a step function inequality

Problem I have been working through content in real analysis and came across across an inequality in a proof that is unclear to me. We are told that a function $\psi$ is a step function on the ...
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Solving Transport Equation with Velocity Switch using Tempered Distributions

Imagine we have the following two transport problems. The first is well known and can be solved using the Fourier transform, but I do not know how to solve the second one. Problem 1: With Constant ...
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Does there exist a sequence satisfying this condition (and what equation would generate it)?

Does there always exist a (finite) sequence $(x_1, \ldots, x_n)$ with $x_n = 1$ and each $x_i \in (x_1, \ldots, x_n)$ is a number $x_i \in \mathbb{Q}$ of the form $x_i = \frac{z}{10}$ for some $z \in \...
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What is $\alpha$? I cannot image $\alpha$ at all. ("Principles of Mathematical Analysis 3rd Edition" by Walter Rudin)

I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin. What is $\alpha$? I can imagine what $\beta(x)=\sum_{n=1}^{N} c_n I(x-s_n)$ is. We can assume that $s_1<s_2&...
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Equation to plot smooth step function for small values?

I'm looking for the simplest equation that will result in a smooth step function (red line in this graph) I have tried looking at sigmoid functions and tried using the method given in this answer. It ...
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Finding integral average form 0 to infinity of a step function

I'm supposed to add a perturbation to $$f_0(v, x, t)= \begin{cases}p(x, t), & v<u(x, t) \\ 0, & v>u(x, t)\end{cases}$$ Getting $ f(x,t,v) = A(x,t) + g(x,t,v)$, where the isotropic part $...
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Cannot solve step function problem in Boas mathematical physics.

I'm trying to solve a problem in Boas(3ed.), Mathematical physics book. Although I put my 3 days to solve it, I couldn't get a solution written on the page. The problem is to show that $\int_{-\infty}^...
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Decomposing Heaviside function with quadratic arguments

I am trying to decompose some step functions with quadratic arguments into the sum of two step functions with linear arguments. If one has the following step function $$\Theta(x^2-y^2),$$ $x$ must ...
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Is it correct to write $ \begin{cases} f(x), & \text{if $x \le d$} \\ \infty, & \text{if $x \gt d$} \end{cases}$ as $f(x) + \infty(x \gt d)$?

I'm reading a book on algorithms and I see they wrote the following step function to minimize the function $f(x)$, where the values of $x$ are infeasible if they are greater than $d$. $$f_{\infty-step}...
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Plotting $θ(x+2)+θ(x+1)+θ(x)x$

The unit step function is given in a combinatory form $θ(x+2)+θ(x+1)+θ(x)x$ , so what we have here is: for $x< 0$: $θ(x+2)=1 \ \ \ when \ \ -2<x$ $θ(x+1)=1 \ \ \ when \ \ -1< x$ $θ(x)x=0 $ ...
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Why is $\epsilon^{*}=\frac{g^{\top} g}{g^{\top} H g}$ the best step size that decreases the 2nd-order Taylor series approx to the function $f (x)$

I don't understand how we determinine the optimal step size that most decreases the Taylor approximation of the second-order Taylor series approximation to the function $f (x)$ around the current ...
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Bounding the difference of step functions

I'm having trouble seeing the justification for what I assume should be a very simple step in a paper's calculations. Let n be a natural number and define $\phi^n_s:=(ns)\wedge1\vee0=\max(\min(ns,1),0)...
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Trying to Solve the Black Scholes PDE with the Green's Function

I have finished the transformation into the Heat Equation. And I am now at the point of establishing the initial conditions. The article I read said the $\max(S-K,0)$ is now the initial condition ...
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Derivative of a multiple of Heaviside step function

First of all, thanks for your time, I have a question. Let's assume that we have: f(x)=(Heaviside's step function)*e^(3x) Now let's assume to calculate the derivate ...
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How to prove $5\mid (6^n -1)$ [duplicate]

How can I prove Inductive Step? $5$ divides $6^n -1$. I've already proved base case and induction Hypothesis but I don't know how to prove Inductive step.
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Find constants $a_i \in \Re$, $b_i \in[0,4]$ such that $\psi(x)$ can be as $\psi(x)= \sum_{i=1}^3 a_ih(x-b_i)$ for $x \in [0,4]$

Let $\psi(x): [0,4] \to \Re$ be the step function defined by $$\psi(x) = \begin{cases} 2, & \text{if $x\in$ [0,1)} \\ -1, & \text{if $x\in$ [1,2)}\\ 1, & \text{if $x\in$ [2,4]} \end{cases}...
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Laplace transform: time scaling of the unit step function

EDIT: It gets the same answer. I have read that the time scaling property of the Laplace transform is not relevant for the unit step function. This property being: $$\mathcal L \left[f(at)\right]= \...
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PDF for a random variable with a step in density at one point (Dirac delta?)

Dirac background and problem statement: The Dirac delta function $\delta(x)$ defined as $$ \delta(x) = \lim_{c \to 0} \delta_c(x) \\ \delta_c(x) = \begin{cases} 1/c && |x| \leq \frac c2 \\ 0 &...
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Proof of integrability of function through step functions

Prove that function $f(x)$ is integrable if and only if for all $ε > 0$, there exist step functions $s, t$ (defined on the same bounds as $f$) such that $s ≤ f ≤ t$ and $$∫(t-s)(x)dx < ε. \quad \...
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Is $e^{1-1/(1-(2x)^{2n})},\,|x|\leq 1/2,\,n>1\,integer$ a "Bump Function"?, also, Could be used to define the rectangular function?

I want to know if this function: for any integer $n>1$ $$ f(t) = e^{1-1/(1-(2x)^{2n})}\cdot(\theta(t+1/2)-\theta(t-1/2))$$ with $\theta(t)$ the unitary step function, which behaves as having: $\...
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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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Integral representation for the heaviside step function

I am studying many-particle quantum theory and I came across the following identity which is used to compute the Fourier transform of Green's functions, $$\theta(\pm \tau) = \mp \lim_{\eta \rightarrow ...
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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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Statistics of Gaussian Random Walk Passed Through the Heaviside Function

Let $$D=\frac{1}{N}\sum_{n=1}^{N}H\left(\xi_{n}-1\right),$$ such that $\xi$ denotes a Gaussian random walk with mean $\mu$ and $\sigma$, passed through the Heaviside function $$H(x-1)=\begin{cases} 1, ...
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a quadratic to approximate a greatest integer function on an interval

I came across this problem: Define $\mathrm{f}: [0,4] \rightarrow \mathbb{R}$ such that $f(x)=[[x]]$ if $x \in [0,4)$ and $f(4)=3$. Find the closest quadratic approximation to $f$ using the set of ...
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Misconception about step function and Heaviside distribution

Given the step function: ${\displaystyle h(x):={\begin{cases}1,&x\geq0\\0,&x<0\end{cases}}}$ How is the Heaviside distribution $H \in {\cal D'}(\mathbb R)$ defined? $ H (f) = h(x) \qquad \...
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Is there a step function with an "in between value" that can be represented by a limit?

I would be interested if there is a function $f(x)$ with the property that when a limit on the parameters of the function is be performed you get a step function with an in between value. What I mean ...
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Smoothing function to combine a polynomial and a constant

click to see the image I have below function: $y=0$ for $x<=0.12$ and $y=80((x/0.12)^8-1)$ for $x>0.12$ I need to smooth out the transition between these two functions. Can I find out the best ...
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Fourier Transform of Heaviside-like functions (with different $t=0$ values)

Consider two functions $x_1(t)$ and $x_2(t)$ as follows: $$ x_1(t)=\left\{\begin{array}{ll} 0 & t<0 \\ 1 & t \geq 0 \end{array}\right. $$ $$ x_2(t)=\left\{\begin{array}{ll} 0 & t\leq 0 \...
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Step functions and Integrability

Definition: A step function $g$ on $[a, b]$ is a bounded function on $[a, b]$ that has only finitely many jump discontinuities at $c_{0}, c_{1}, c_{2}, \ldots, c_{k}$ in $[a, b]$ and is constant on ...
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What's an expression for the function of a limit?

If we define the Heaviside step function H(x) in limit notation, as per below, this yields 1/2 at ...
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Integrability of a function defined using an enumeration of the rationals in $[0,1)$

We have $(r_k),k\geq1$ as an enumeration for the rationals in $[0,1)$ and $f$ is a function defined as $$\large f=\sum_{k\geq 1}\frac{1}{k^2}\mathbf{1}_{[0,r_k)}$$ where $\mathbf{1}$ is the indicator ...
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Simple step function integration

I have a function defined by $f(x)=n$ for $n-1\le x \lt n$ for $n\in {1,...100}$. With $f(x)=0$ elsewhere. I am now trying to find $\int f$. My first though would be that $\int f=\sum_{i=1}^{100} i=...
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Proving $\int_0^a x^b dx=\frac{a^{b+1}}{b+1}$

I want to show that $\int_0^a x^b dx=\frac{a^{b+1}}{b+1}$. Whereby $a,b\in\mathbb{N}$. I know this isn't too hard to prove with the power rule etc however I would like to form a proof using step ...
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Closed form expression for $\sum_{n=0}^\infty H(x - n y) z^n$ involving Heaviside functions

For $x>0$, $y>0$ and $0 < z < 1$ consider the sum $$ \sum_{n=0}^\infty H(x - n y) z^n\ , $$ where $H$ is the Heaviside step function. Is there a way to write down a closed form expression ...
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Definite integral with derivative of Heaviside function

I'm working on a problem I have been dealing with unsuccessfully for months now, so any help is greatly appreciated! The context I have to solve an integral of the form $$\begin{align} \int_0^{\infty}...
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Function of the sigma (sum symbol) in the definition of a simple function?

Simple functions assume finitely many values in their image, and can be written as $$f(\omega)= \sum_{i=1}^n a_i \mathbb I_{A_i}(\omega), \quad \forall \omega \in \Omega$$ where $a_i \geq 0, \forall ...
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2 answers
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Derivative of $\mbox{sgn}$

I get a different result than the book I'm reading for the derivative of the sign function. Let's define the sign function, $x \mapsto \mbox{sgn}(x)$, as $$ \mbox{sgn} (x) = \begin{cases} 1 &...
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Need some help in understanding a step-function notation

So I was reading this paper and I saw this nonsmooth nonlinear function. But, I am not familiar with this kind of notation, does it denote some kind of a step-function with the value $1$ if $X \leq a$ ...
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Is this function $f(x,y)$ regulated when fixing $x$ and then when fixing $y$?

I have come across this question when studying for my exams. I have gotten somewhere but I am struggling coming up with solid reasons on whether it is regulated or not. The function is $f:[0,1]^2 \to [...
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Find integral limits for step functions

I have an equation of the form $$f(t) + \int_0^t H(t') dt' = c$$. For now, assume that $f(t)$ is linear, and $H(t)$ is some arbitrary step function (not necessarily just a standard heaviside step ...
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What if we take step functions instead of simple functions in the Lebesgue integral [duplicate]

When we define the Lebesgue integral, we first define it for simple functions $s(x) = \sum\limits_{j=1}^n c_j\chi_{A_j}(x)$ (where $A_j$ are measurable) as $\int sd\mu = \sum\limits_{i=j}^n c_j \mu(...
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Can someone find $\vec{A}$ for this example [found in my TB : Griffith] with this method?

Example 10.2 of 3rd edition Griffith [electrodynamics] click here to read this question So I thought to convert I into $\vec{J}$ as follows : $$\vec{J}(\vec{r},t)= I_o\theta(t)\delta(x)\delta(y)\hat{z}...
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Is $u(t)=u(2t)$ true?

Since $u(t)=1$ for $t>0$ and $0$ otherwise, this means that $u(t)=u(2t)$ for all $t$. Also, they both have identical graphs. But when obtaining the derivative of both of these functions they give ...
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Does $t\delta(t) = 0$?

In middle of solving a problem I encountered terms looking like this: $$t\delta(t)$$ Or in a more general form: $$(t-t_{0})\delta(t-t_{0})$$ It appears that in the textbook I am reading "Linear ...
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Find the limit of a step function with two different variables

Given the following step function: $$ f_n(x) = \left\{ \begin{array}{ll} 0 & \quad 0 \leq x < 1 - \frac{1}{n} \\ n & \quad 1 - \frac{1}{n} \leq x < 1 \\ ...
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Query about a proof of Riemann integrability of step/simple functions

Let $f:[a,b]\to\Bbb{R}$ be a step function is of the form $$f=a_11_{[a,t_1]}+\sum\limits_{i=2}^n a_{i} 1_{(t_{i-1},t_{i}]}$$ i.e. in simple words $f(x)=a_1$ for all $x\in [a,t_1]$ and $f(x)=a_{i}$ for ...
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