Questions tagged [piecewise-continuity]

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

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sum of piecewise functions - twice continously differentiable?

Let f be a cubic interpolant. I want to express it as a sum of "basis functions" multipled by interpolation values. That is, $$ f(x) = \sum\limits_{i=1}^{n-1} S_i(x)f_i $$ where $f_i$ are ...
Simon's user avatar
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How to get a Filippov solution?

Recently, I read a book ISSN 2195-9862 about the Filippov theory. There is a differential inclusion $$\dot{x}\in F(x)=\begin{cases} -1&x>0\\ [-1,1]&x=0\\ 1&x<0 \end{cases}\\ x(t_0)=...
Liu C's user avatar
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Asymptotic expansion of a special peicewise function

My professor challenged me to determine an asymptotic expansion of this piecewise function : $$f(x) = \left\{ \begin{array}{c} xsin(1/x) & if & x \in \mathbb{Q} & \\ \\ x & if & ...
Salem's user avatar
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Existence of antiderivative for a piecewise function

How to determine the value of an $a \in \mathbb{R}$ parameter so that there exists an antiderivative for the $f:\mathbb{R} \Rightarrow \mathbb{R}$, $$f(x)=\begin{cases}\sin\frac{e^x}{x},& x\neq 0\\...
Birgitt's user avatar
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Differentiabiliy of discontinous function

I was watching some video about differentiability at some point and I was presented with this function: $f(x)=\left\{ \begin{aligned} \frac{1}{2}x,\quad x < 1\\ \sqrt{x}-1 ,\quad x\ge1\\ \end{...
Mire's user avatar
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Piecewise to Heaviside

I really need help with this exercise, I try by days. Use Heaviside in this exercise: $$f(t) = \begin{cases} 2t^2 + 4t +1 ,& 0 \leq t < 6 \\4t^2 -16t+50 ,& t \geq 6\end{cases}$$ for show L{...
Estre's user avatar
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Step functions vs piecewise continiuty

Definition: A function $f :[a, b] \Rightarrow\mathbb{R}$ is said to be piece-wise continuous if there exists a partition $a = t_{0} < t_{1} <...<t_m =b $, such that for each $k$, the ...
emil agazade's user avatar
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cadlag functions equal a.e. implies they are equal everywhere

Let $f,g:\mathbb R\to\mathbb R$ be cadlag (continuous from the right and limits existing from the left) functions that are equal almost everywhere with respect to Lebesgue measure. I.e. $f(t+)=f(t)$ ...
jdods's user avatar
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Composition of continuous and piecewise $C^{\infty}$ functions.

Let $f:[a,b]\rightarrow[c,d]$ be a continuous and piecewise $C^{\infty}$ function sending $a$ to $c$ and $b$ to $d$ and let $g:[c,d]\rightarrow \mathbb{R}$ be a real continuous and piecewise $C^{\...
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Demonstrating Piecewise Linearity in a Parametrized Optimization Solution

Let $\mathbf{B}$ be a definite positive square matrix of size $n \times n$, and $\mathbf{b}$ an $n$-sized vector. It can be shown that the solution of $\arg\min_x \left(\mathbf{x}^T \mathbf{B} \mathbf{...
cyril's user avatar
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Formal way to prove discontinuity of function

Say I want to show the function f(x)=x+1 when x is rational and f(x)=2x when x is irrational is only continuous at 1. Clearly, it is only continuous when x=1. To show this, I proved $\lim_{x \to 1} f(...
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Proving continuity of piecewise function separated by rationals and irrationals

I want to prove where the function f(x)=x when x is rational and f(x)=2x-1 when x is irrational is continuous. Clearly, it is only continuous when x=1. To show this, I proved $\lim_{x \to 1} f(x) = f(...
user124820929's user avatar
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Find a constant such that the function is differentiable everywhere.

I am doing some practice problems and I need to find a constant $a$ such that $$ f(x) = \begin{cases} \hfill x\left(1+2x\sin\left(\frac{1}{x}\right)\right) \hfill & \text{ $x \neq 0$} ...
Newbie1000's user avatar
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Why is a derivative undefined at its discontinuities?

This question deals with why the derivative of $f$ is not defined at discontinuities in $f$. I found the answers satisfactory. My question deals with why the derivative is not defined at ...
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Piecewise function outputs the sum of the digits in base 3 representation

Recently, I encontered a rather interesting function in the solution booklet to a math contest. It is a piecewise function that outputs the sum of the digits of the input in base $3$. Could anyone ...
Eddie Wang's user avatar
2 votes
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60 views

Any known discontinuous functions which show similar behaviour

I come from a physics background, and as a result of one of my simulations, I get the following function: On the x axis, the sampled points are of the form $m/n$ with $n \in [1,20]$ and $m \in {1,2,.....
YeatTheorem's user avatar
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Analyzing Continuity of a Piecewise Function and Addressing Discontinuity at the Excluded Point

This year, in one of the exercises for university admission in the subject of Mathematics in Spain, it states the following: Given the real function of a real variable defined over its domain as $$f(x)...
Jose Maria Lopez Belinchon's user avatar
2 votes
1 answer
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Finding the derivative of a function with two absolute values within it, using a piecewise function

I'm trying to solve some a problem relating to absolute values. I found online the strategy for solving similar functions from here: https://www.youtube.com/watch?v=eIHtq67nh7w&list=...
mintteaplease's user avatar
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How do I calculate the values for $W$, $X$, $Y$, and $Z$ in this project, given specific constraints?

I am exploring the math of smooth motion. I have everything sorted out for zero through four degrees of smoothness, but I'm stuck on some of the transition-time values for the fifth degree of ...
Lawton's user avatar
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$f(0) = 0$, $f(1) = 1$ and $f(n)=f(n-T_{m-1})-f(T_m-n)$ if $T_m:=\frac{m(m+1)}2\ge n>T_{m-1}.$ Find the smallest $n$ such that $f(n) = 4.$

Here's the question: For nonnegative integer $n$, the following are true: $$ \text{(a) } f(0) = 0$$ $$ \text{(b) } f(1) = 1$$ $$ \text{(c) } f(n) = f(n - \frac{m(m - 1)}{2}) - f(\frac{m(m + 1) }{2} - ...
codexistent's user avatar
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Complete with continuity a piecewise function

Good morning, I have been working with piecewise functions lately. I am trying to completethis piecewise function, in order to make it smooth, with both first and second derivative smooth as well (...
Marco Lugarà's user avatar
1 vote
1 answer
75 views

How to smoothly remove boundary discontinuities in functions and their derivatives at the edges of their defined domain?

If I want to Fourier transform a function $$t \in [-1,1] \to f $$ but this function and it's first $n$ differentials are not equal at the edges : $$f^{(k)}(-1) \neq f^{(k)}(1) , k \in \{0,1,\cdots,n\}$...
mathreadler's user avatar
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finding the values of a and b(real numbers) that makes the function differentiable at any point of its domain

I'm given this function: $$ f(x) = \begin{cases} (lnx)^4 & 0<x< e \\ ax+b & x≥e \end{cases} $$ and I'm asked for which real values of 'a' and 'b' the function is differentiable at any ...
Ofri K's user avatar
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Continuity of solutions of Laplace's equation on the unit disk with piecewise $\mathcal{C}^1$ boundary condition

The question is the following: Let $f$ be a piecewise $\mathcal{C}^1(\mathbb{R})$, $2\pi$-periodic function, and let $a_{n}[f],b_{n}[f]$ be its real Fourier coefficients. Show that the series \begin{...
Mario's user avatar
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Is a piecewise linear function always a sum of concave and convex functions?

If I take a piecewise linear function (piecewise affine) is it true that I can always write it as a sum of concave and convex functions? My understanding of this page https://mjo.osborne.economics....
robotsheepboy's user avatar
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Second Partial Derivative of a piecewise function at point (0, 0)

I know how to get the first partial derivative of a piecewise function at a certain point but I'm unsure how to get the second one. I'm trying to do it for the following function: $f(x,y) = \begin{...
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1 answer
306 views

Help with piecewise function?

I was hoping someone could help me in finding a solution to a question regarding piecewise functions? I'm struggling because everything I have read / learned only has a single pronumeral and I can't ...
Glenn's user avatar
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Question on partitions of piecewise functions.

I have here three step-wise functions, the first two in the image below, $f\, and \ g$, have four intervals, while their sum, $f+g=h$ has 8 intervals. The first two are defined respectively on, $f: [-...
Superunknown's user avatar
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How do you rewrite a piecewise function using dirac delta

I need to solve a differential, but a portion of the problem is written as a piecewise function with 4 parts, where one part is equal to infinity. I am not sure how to write this using dirac delta. I ...
Samantha Garcia's user avatar
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Fourier Series Transformation of Piecewise Function

I would appreciate some help with the question below. It is just for personal study to get ahead for next year, but the textbook doesn't have answers, and I am unsure how to do b and d specifically. I ...
HSC Coach's user avatar
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Constructing a piecewise function with a constant area under the curve for all pieces

There is a list of times corresponding to flashes of the status indicator LED of an electric power meter. The LED flashes when 1 Wh of energy (E) has been used. Because energy is the product of power (...
kgello's user avatar
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0 votes
1 answer
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Proving that a specific function is discontinuous on a specific curve

I need to investigate at which points the following function is continuous: It is clear that for any point with $y \neq x^{2} $, it is continuous. I suspect that along the curve $y =x^{2} $ it is not ...
autodidacti's user avatar
1 vote
0 answers
81 views

Find the values of $\alpha$ such that the function is continuous

Given the function $$f_a(x,y)=\begin{cases}y^\alpha \frac{\sin(xy)}{x^2+y^2}\ \ \ \ \text{ for }(x,y)\neq (0,0)\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ for }(x,y)=(0,0)\end{cases}$$ with $\alpha=...
Aley20's user avatar
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7 votes
1 answer
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Is there a good way to simplify this expression?

The Short Version Is there a way to simplify this expression? $$ \left(\left(\left(d × \left(\frac{j}{2}\right)^2\right)^2 − \frac{1}{27} × \left(\frac{j^2}{2 s}\right)^6\right)^\frac{1}{2} + d × \...
Lawton's user avatar
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0 votes
1 answer
38 views

Question over the writing of a function

This is a really silly question, I know, but I just wanted to know if I'm too "strict" or if it's really an error. An exercise our professor given us start with this: be $f$ a function ...
Heidegger's user avatar
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1 vote
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If $f'(t)=x'(t)\ \theta(T-t)$, What is the accurate formulation of $f(t)$? (antiderivative over distributions/special_functions)

If $f'(t)=x'(t)\ \theta(T-t)$, What is the accurate formulation of $f(t)$? (antiderivative over distributions/special_functions) Maybe the question look simple, by I need to understand Why it is as so,...
Joako's user avatar
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1 vote
1 answer
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Are these two piecewise functions equivalent or different?

I got this solution to the inequality $|x - 5| + |x - 4| \ge 3:$ $$|x - 5| + |x - 4| - 3 = \begin{cases} 2x - 12& \text{if }x > 5\\ -2 & \text{if }4 \le x \le 5\\ -2x + 6& \text{if }x &...
Oofy2000's user avatar
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4 votes
2 answers
329 views

Conflicting results for $\int\frac{|x+1|}{x}\,\mathrm dx$

I've encountered an obstacle while trying to solve the following integral: $$\int{\frac{\sqrt{x^2+2x+1}}{x}dx}$$ First thing we shall do is see that under the square root is actually $(x+1)^2$. When I ...
bb_823's user avatar
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2 votes
2 answers
123 views

Piecewise limit does not exist proof [closed]

Let $$f(x)=\begin{cases} 1&\text{if }x\leq 0\\ -1&\text{if }x>0 \end{cases}$$ Prove, using the precise definition of the limit, that $\lim\limits_{x\rightarrow 0} f(x)$ does not exist. I ...
squito's user avatar
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3 votes
1 answer
177 views

Why would the |f(x)| be non differentiable for x belonging to Real Numbers?

I'm just started calculus and come across a statement in 1 of my books (VG Advanced Problems in Mathematics) that goes like: If $y = f(x)$ is differentiable for x belonging to the set of Real numbers, ...
Elizabeth Huffman's user avatar
0 votes
1 answer
306 views

Discontinuities of Piecewise functions with undefined points

In multiple class examples, and internet examples on discontinuities. I often see that the undefined points are often called "the points at which the function is discontinuous". So If I have ...
RMS's user avatar
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1 answer
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Indefinite as sumand

My question is based on some questions about piecewise functions. The problem is when I "sum" two piecewise functions that don't share any interval. When I draw the graphic of this function ...
Sickman6661's user avatar
2 votes
0 answers
45 views

Fourier Series Derivative of saw tooth-like function

So I have this periodic signal x(t), $$x(t)=\begin{cases} 5t & \text{ if } x \in [0.2,0.4]\\ -10t & \text{ if } x \in [0.4,0.5] \\ 0 & \text{ if } x \in [0.5, 0.6] \end{cases}$$ and I was ...
arpg's user avatar
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0 answers
24 views

Calculate ${f^{\ast}}'(0)$ if $f^{\ast}$ is an odd expansion of the function f which is defined on the interval $0\leq x\leq L$

Imagine that the function $f(x)$ is defined on the interval $0\leq x\leq L$. $f$ is defined and continuous on $[0,L]$ and differentiable on $(0,L)$. also $f(0)=f(L)=0$ (The function $f$ is the initial ...
absolutezero's user avatar
5 votes
1 answer
108 views

Proving discontinuity at $\,x =1\,$ using $\,\varepsilon$-$\delta\,$ definition

I was doing a proof for discontinuity using $\,\varepsilon$-$\delta\,$ definition but I’m not sure whether the proof is right. Would you mind checking it for me please, thanks! $g(x)=\begin{cases}\...
a22's user avatar
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0 votes
1 answer
121 views

The number of disjoint intervals over which the function $f(x) = |0.5x^2−| x | |$ is decreasing is

The number of disjoint intervals over which the function $f(x) = |0.5x^ 2−| x | |$ is decreasing is A)one B)two C)three D) none of these I actually solved it and the answer is three but I want to ...
sachin's user avatar
  • 13
2 votes
1 answer
51 views

Most efficient way to find the "solution intervals" / "case conditions"? (problem regarding a function with absolute value notation):

I am trying to write the function $f(x) = |x^2-1|+|x|-1$ without the notation for the absolute value. "||" It makes logical sense to consider four cases, because each term in the absolute ...
JosefS's user avatar
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4 votes
1 answer
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Need help choosing delta to prove continuity for $f(x)=\frac1x$ at $x=\frac12$ in a piecewise function.

I am trying to prove continuity of the function $$\text{Let } f(x)=\begin{cases} f_1(x) = \frac{1}{x} & \text{if }x\in (0,\frac{1}{2}] \\ f_2(x) = 2 & \text{if }x\in(\frac{1}{2},1)...
DoubleV's user avatar
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0 votes
0 answers
393 views

Fourier Transform of cosine in the interval $-\frac{\pi}{2}$ to $\frac{\pi}{2}$

Calculate the Fourier transform of the function: \begin{equation} f(x) = \begin{cases}\cos(x) & \text{ if } \frac{-\pi}{2}\leq x \leq \frac{\pi}{2}\\ 0 &\text{ otherwise.}\end{cases}\end{...
Zeeko's user avatar
  • 107
1 vote
2 answers
63 views

Proving that a piecewise function is continuous for all reals except one point

Im having a bit of trouble proving a function is continuous for all reals except for one point. I can do the proof for one point but I'm having trouble extending the proof to all reals except one ...
non ducor duco's user avatar

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