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 real analysis and measure theory to approximate integrable functions.

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Riemann integrability for step function

Here is the problem: Fix $c\in\mathbb{R}$ and define $g:[0,2]\to\mathbb{R}$ by $$g(x)=\begin{cases}2 &\text{if } 0\le x<1\\c &\text{if } x=1\\ 1&\text{if } 1< x\le 2.\\\end{cases}$$ ...
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Define this step function over the rational numbers

In desmos I plotted a step function (I only plotted 30% of it). Here is my graph: This function is a function from $\Bbb Q\cap (0,1) \to \Bbb Q\cap(0,1).$ The step function is generated by counting ...
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The product of a Heaviside function and Dirac function centered around different points.

Let $a,b$ be two real-numbers such that $a \neq b$. Let $\iota(x \leq a)$ be the Heaviside step function in variable $x$ and $\delta_b(x)$ be the Dirac Delta function centered around $b$. I have two ...
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Evaluating the integral $\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$

I want to evaluate the following integral $$\int_0^{1}\int_{0}^{2\pi}x\delta (v-x\cos\theta)dxd\theta$$ According to the book where I found the exercise, the answer is $$C\Theta(1-v^2)\sqrt{1-v^2}$$ ...
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Laplace Transform of a Piece-wise function with a Weibull distribution.

Suppose I have the following piecewise function: $$Q(t) = \begin{cases} W(t) & t<T \\ 1 & t=T \\ 0 & t>T \end{cases}$$ ...
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Easy Lebesgue integral of a non-horizontal line but by definition

Maybe a dumb question based on all the questions I've asked for the last decade, but what's the general way to do the Lebesgue integral of some non-negative (measurable?) function that is Riemann ...
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Exponencial grow, with controlable outcome and steps.

I'm not a mathematician so I'm not sure how to calculate what I need. The problem is, I want to go from 0 to 0.9, growing exponentially, over a determined amount of steps. And with every step added ...
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Is the integral of the Dirac delta function equal to the integral of the Dirac delta function times the Heavisde unit step function? [duplicate]

Given that the Dirac delta function is defined as: $$\delta(t) = \begin{cases} +\infty, & t = 0\\[2ex] 0, & t \neq 0\\[2ex] \end{cases}$$ And that the Heaviside unit step function is defined ...
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Maximum of $y''$ for BVP with $y''''\leq0$.

Consider the following boundary value problem for some $L>0$ and $w(x)\geq 0$: $$\frac{d^4y}{dx^4}=-w(x)\,;\,\,\,y(0)=y(L)=0,\,y'(0)=y'(L)=0.$$ Here $w$ can be pathological: a step-function or an ...
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