Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [piecewise-continuity]

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

0
votes
0answers
17 views

Solve differential equation of piecewise function Matlab.

I have a piecewise function called $y(x)$ and another function called $v(x)$ which is a derivative of $y(t)$. Now I need to solve this function and plot in Matlab. To do this I've used this code and ...
0
votes
1answer
36 views

For what values of $p $ and $q$, this function is continuous?

I am not sure how to begin solving this. With two variables $p$ and $q$ that need to be solved for in order for all three to be continuous. $$ f(x) = \left\{ \begin{array}{rl} x+p &\...
0
votes
1answer
51 views

Proof that a piecewise function is uniformly continuous

Let the function $f : \Bbb{R} \to \Bbb{R}$ be denoted by: $$f(x) = \begin{cases} \dfrac13 \sin(3x) + x^4 \sin\dfrac{1}{x^3}, \quad & x \ne 0 \\ 0, & x = 0 \end{cases}$$ Prove that $f$ is ...
1
vote
1answer
33 views

Fundamental Theorem of Calculus Piecewise Function help?

Let: $$f(x)= \begin{cases} 0, & x < -4 \\ 5, \qquad\quad& \llap{-4 \le{}} x < -1 \\ -2, & \llap{-1 \le{}} x < 3 \\ 0,& x \ge 3\end{cases}$$ $$g(x) = \int_{-4}^x f(t)...
3
votes
3answers
72 views

If $f(x,y)=9-x^2-y^2$ if $x^2+y^2\leq9$ and $f(x,y)=0$ if $x^2+y^2>9$ study what happens at $(3,0)$

If$$f(x,y)=\begin{cases}9-x^2-y^2&\text{if }x^2+y^2\leq9\\0&\text{if }x^2+y^2>9\end{cases}$$study the continuity and existence of partial derivative with respect to $y$ at point $(3,0)$. ...
0
votes
0answers
18 views

Left continuity of $\mathrm{1}_{c<x}+x \mathrm{1}_{x\leq c}$?

I'm currently reading an argument that I'm convinced is incorrect and am asking here to see if error is mine or my source's. Simplifying the function somewhat, the argument claims that the function $\...
1
vote
1answer
37 views

Inverse of a piecewise continuous function.

I am trying to find the inverse of a two dimensional map $f\left(\begin{bmatrix} x\\y\end{bmatrix}\right)$, For example $$ f\left(\begin{bmatrix}x\\y \end{bmatrix}\right) = \begin{cases} ax + by, &...
0
votes
1answer
28 views

Does piecewise continuous imply Borel measurable?

It's extremely well-known that continuous functions are Borel measurable, but what about piecewise continuous functions? For the Lebesgue measure, I suspect that we'd have a proof as simple as "...
0
votes
1answer
39 views

Jerk-controlled minimal time second-order velocity smoothing problem.

Kinematic profile: $T1$ - increasing acceleration time (max jerk) $T2$ - constant acceleration time (zero jerk) $T3$ - decreasing acceleration time (-max jerk) Jerk, acceleration, velocity The ...
-2
votes
1answer
63 views

Proving piecewise function is not continuous [closed]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = \left\{ \begin{array}{ll} 2x & \quad x \text{ is rational} \\ -2x & \quad x \text{ is irrational }...
0
votes
1answer
30 views

limit of multivariable piecewise function(calculus)

I got a piecewise function defined as $$f(x,y)= \begin{cases} 1, & 0<y<x^2\\ 0, &\quad\; \;\text{else} \end{cases}$$ How to find $\lim_{(x,y) \to (0,0)}f(x,y)$ along the $y=mx$ path ? ...
1
vote
1answer
20 views

Properties of piecewise functions and their derivatives: summation and integration

I have the following functions $g(x) = 16x^2(1-x)^2$ on $[0,1]$ $f(x) = 1 - g(x)$ on $[0,0.5]$ and $f(x) = 0$ on $[0.5,1]$ $h(x) = 0$ on $[0,0.5]$ and $h(x) = 1 - g(x)$ on $[0.5,1]$ As one can ...
1
vote
1answer
34 views

how do I find the limit

$\lim_{x\to1^+}f(x)$, $\lim_{x\to1^-} f(x)$, and $\lim_{x\to1} f(x)$, where $f(x)= \begin{cases} 2x+1 & \mbox{if } x≤1 \\ 4-x^2 & \mbox{if } x>1 \end{cases}$ I do not understand how to ...
1
vote
3answers
14 views

Checking for Continuous function

I have the below question, i understand how to check for continuity for individual equations in a piecewise function but i dont understand how to find continuity for the whole function with only 1 ...
4
votes
2answers
101 views

Why is it necessary to split the definite integral of a piecewise function into a sum

The second fundamental theorem of calculus (Newton-Leibniz) tells us that: If $f$ is a real-valued function on a closed interval $[a, b]$ and $F$ is an antiderivative of $f$ in $[a,b]$ s.t. $F'(x)=f(...
0
votes
1answer
47 views

How can I graph this derivative of a quarter of a semicircle?

For graphing the derivative of the circle, I know that the equation of a circle is $x^2+y^2 = r^2$ and in this case r = 4 With implicit differentiation I know that $y' = \frac{-x}{y}$ or $\frac{-x}{\...
3
votes
0answers
22 views

how to graph the derivatives of certain kinds of piecewise functions

For questions like these, how can I graph the derivative? For the first image, the sideways x^3 graph is the original and for the second image the v-shaped thing is the original function. For the ...
4
votes
1answer
73 views

How can I graph the derivative of 1/4th of a circle or a semicircle in a piecewise function? (Also other kinds of piecewise functions)

I'm having trouble with questions like these. In the first image, the original function is what is the two sharp lines and a semicircle in between. I understand how to find and graph the derivative of ...
0
votes
0answers
32 views

Integral and inverse CDF of piece-wise Gaussian distortion in polar coordinates

I have a series of radii which are used to describe a series of concentric rings. I also have a piece-wise function which is evaluated on every radii. The function resembles a Gaussian-like distortion ...
2
votes
1answer
104 views

How to fit data to a piecewise function?

My question today regards a set of data that I wish to fit a piecewise-defined continuous function. This data set covers a domain of x-values from $0$ to $\mu$ on the x-axis. What I need is to ...
2
votes
1answer
35 views

First and second differentiability of the piecewise function $x^4\sin(\frac{1}{x})$ if $x \neq 0$ and $0$ if $x=0$

I have the following function $f(x)$ defined as $x^4\sin(\frac{1}{x})$ if $x \neq 0$ and $0$ if $x=0$. And I'm asked if the function is: a) differentiable b) two times differentiable c) two times ...
3
votes
1answer
39 views

Did I make a mistake when finding the intervals this function is continous on?

I was given a function f given by $f(x) = \begin{cases} \frac{x^2 - a^2}{x - a \;} \textit{if} \; x \neq a, \\ 2a \; \textit{if} \; x=a \\ \end{cases}$, and told to find the intervals over which $f$ ...
0
votes
0answers
19 views

$f\in C^1$ piecewise: why should the derivative be finite on the borders of the pieces?

$f\in C^1([a,b];\Bbb R)$ piecewise $\iff\ \exists\sigma= \{a=a_0<a_1<\dots<a_n=b\}$ a partition of $[a,b]\text{ such that } f'|_{(a_i,a_{i+1})}\in C^0$ $\forall i\in\{0,\dots,n-1\}$ and $f\...
1
vote
1answer
28 views

Functions that Tend To Non-Smooth Functions as Some Parameter Tends to Infinity

I recently saw a post in which the query was about a function that tends to the Dirac delta function as a parameter in it tends to infinity. The function chosen was $${(1+\cos x)^n\over C}$$ as $n\to\...
1
vote
1answer
102 views

Plotting a solution of a differential equation with Sagemath

I need to solve a differential equation. The solution will depend on $t$ and $q$, and I need to define that $q$ piecewise depending on $t$. ...
0
votes
3answers
30 views

Need some help with continuity for the following piece-wise functions

Let $$f(x)=\begin{cases}x^2−7 &x\leq c\\10x−32 &x>c \end{cases}$$ If $f(x)$ is continuous everywhere, then what is $c$ equal to? Also, where is $$f(x)=\begin{cases}x+2 &x<0\\e^x &...
8
votes
2answers
1k views

If a function is continuous everywhere, but undefined at one point, is it still continuous?

This is a question regarding the definition of continuity. My understanding of continuity is that a function is continuous at a point when it holds that $$\lim_{x\to a^-}f(x) = f(a) = \lim_{x\to a^+}...
5
votes
0answers
90 views

Proving a definite integral is finite

I have a integral which I have to prove is finite. $$\int_{-\pi }^{\pi } \left(\frac{x \cos x-\sin x}{x^2}\right)^2 dx $$ call the function inside $g(x)$, where $g(x) = (f'(x))^2$ and where $f(x) =...
2
votes
3answers
62 views

Showing differentiability of $g(x)=\begin{cases}\frac{f(x)}{x},&\text{$x\neq0$}\\f'(0),&x=0\end{cases}$ given that $f(0)=0$

Let $f$ be a twice-differentiable function of $\mathbb R$ with $f(0)=0$. Define $$g(x)=\begin{cases}\frac{f(x)}{x},&\text{$x\neq0$}\\f'(0),&x=0\end{cases}$$ Prove that $g$ is a differentiable ...
1
vote
1answer
16 views

Differentiability of a piecewise (salt-pepper like) function

Today I ran into the following problem: Let $f$ be a function such that $$ f(x) = \begin{cases} \frac{1}{4^n}, &x=\frac{1}{2^n}, \;\; n\in \{1,2,3,\ldots\} \\ 0, &x\neq\frac{1}{2^n} \end{...
0
votes
1answer
67 views

How can I make this two parameter piecewise function continuous?

$ f(x)= \begin{cases} e^{a+bx}&\text{ }\, 0\leq x\ < \frac{1}{2}\\ \mu &\text{ }\, \frac{1}{2} \leq x\leq 1\\ \end{cases} $ ASSUME $\mu$ is a constant and a and b are the two parameters ...
0
votes
1answer
66 views

Measurability of piecewise constant function

If $(X, \mathbb A, m)$ is a measurable space and if we have a function $f: (X, \mathbb A) \to (\mathbb R, B(\mathbb R))$ which is measurable, non-negative and piecewise constant, why does - in this ...
0
votes
0answers
23 views

Continuity of a piecewise function (rarefaction wave for quasilinear pde's) - question on domain

In studying elementary theory of PDE's from Salsa, the author talks about weak solution for conservation laws (quasilinear first order PDE). In particular, the Cauchy problem $$\begin{cases} \rho_t+...
0
votes
1answer
21 views

Proving A Function Is Continuous On Interval

we have a huge Real Analysis exam on Monday, I understand the idea behind continuous functions where the interval is specified, i.e. [a,b] on a single function, but this question confuses me as it ...
0
votes
0answers
29 views

Differentiability of a Piecewise Function $f(x)$ based on the continuity of $f(x)$ and $f'(x)$

I know that, in order for a function $f(x)$ to be differentiable at a value $x=a$, the derivative $f'(a)$ must exist, i.e. $\lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$ must exist. So, one would use ...
4
votes
1answer
45 views

Computation of piece-wise linear hat functions

I have a discretized 3D surface for which I want to compute piece-wise linear hat functions. I assumed these functions are of the following form: $$\phi = ax + by + cz + d$$ with the property of $\...
0
votes
1answer
54 views

Derivative of multivariable piecewise function

I want to know how I can make the derivative of this piecewise function respect to the X variable. I know that in the point (0,0) you have to use the definition but I need the general derivative of ...
3
votes
3answers
26 views

Piecewise function and differentiation quotient

We have function $y = f(x) = |2x + 1|$ Can we simply say that $|2x + 1|' = 2$? No because if we use the chain rule then we get $\left(\left|2x+1\right|\right)'\:=\frac{2\left(2x+1\right)}{\left|2x+1\...
0
votes
1answer
72 views

What is the mathematical approach of inversing a function resulting in a piece-wise solution?

I've been trying to find the inverse of $$f(x) = e^{-\left(\displaystyle \frac{x}{\sqrt{1-x^2}}\right) \displaystyle \pi }$$ Here are my steps $$ \begin{align} x & = e^{-\left(\displaystyle \frac{...
1
vote
1answer
49 views

Determine where this function is continuous - floor function

Determine where this function is continuous: $$f(x)= \begin{cases} \frac{1}{x-\lfloor x\rfloor}, & \text{for } x \notin \mathbb{Z}\\ 1, & \text{for } x \in \mathbb{Z}\\ \end{cases} $$ ...
1
vote
1answer
134 views

How to rigorously show that maximum of linear functions is piecewise linear?

Note: This is an extremely basic question, which is why the fact I can't figure out the answer easily concerns me. This problem is from Boyd and Vandenberghe. Consider $f: \mathbb{R} \to \mathbb{R}$...
0
votes
1answer
26 views

Unsure why I can't find a limit of PDF using CASIO Classpad with Piecewise function [closed]

I am trying to find the value of 'd' using the CASIO Classpad. I have been given a hybrid (piecewise) function and, having defined it on the calculator, I need to know the value of 'd' which will give ...
-1
votes
1answer
33 views

Doubt on piecewise function

Suppose we have a function given by: $f(x)= \begin{cases} x & x < 0\\ x+1 & x \ge 0 \end{cases}$ Then $f(|x|)= \begin{cases} |x| & x < 0 \\ |x|+1 & x \ge 0\end{cases}$ Or $...
-1
votes
1answer
53 views

How can I find $\delta$ given $\epsilon$?

How do I need to show that there exist $\delta>0$ corresponding to $\epsilon= 0.03$, when $(x,y,z)\to (0,0,0)$ using $\epsilon$-$\delta$ definition of limit. Given that $f(x,y,z)$ is $$f(x,y,z)=\...
0
votes
0answers
55 views

Expanding the Piecewise function at $(0,1;0)$ through Taylor series?

$f_{r}(x,y;\xi) = f_{X} (x,y;\xi) $ when $y \leq g(x;\xi)$ and $f_{r}(x,y;\xi) = f_{Y} (x,y;\xi)$ when $y \geq g(x;\xi)$. where $g(x;\xi) = 1 - p_{1}x - p_{3}\xi + O(x^2 + \xi^2)$ and $\xi$ is a real ...
4
votes
1answer
55 views

Discontinuity - Unsure If Piecewise Equation(s) Have Them

I have a question on whether the functions following have a discontinuity, and if not, what are the points where two functions meet. First, the piece wise equation : \begin{align*} f(x)= \begin{...
1
vote
0answers
44 views

Piecewise smooth function , Jacobian and locally invertible?

Say $f$ is a piecewise smooth function that is $f(x,y;\chi)=f_{1}(x,y;\chi)$ if $y\leq g(x;\chi)$ and $f(x,y;\chi)=f_{2}(x,y;\chi)$ if $y \geq g(x;\chi)$ where $g$ is a $C^2$ function and $\chi$ a ...
0
votes
1answer
227 views

Signum if x tends to 0

$$\lim_{x\to0} Sgn(x)$$ What should its value be? I know $Sgn(0)=0$, but if we imply that x tends to 0, shouldn't it be an infinitesimal number close to 0, but not equal to zero, and shouldn't its ...
0
votes
3answers
180 views

How to maximize a piecewise linear convex function $f: \mathbb{R}^n\to \mathbb{R}$?

How to maximize a piecewise linear convex function $f: \mathbb{R}^n\to \mathbb{R}$? I can see that there are many references for minimizing a piecewise linear convex function but not maximizing such a ...
0
votes
0answers
49 views

Does piecewise continuity of $f'$ implies that the discontinuities of $f$ are jump discontinuities?

In this pdf, the following theorem is stated without proof: I'm not sure how this mathematically is accurate. My question is: Do the limits $f(x^+)$ and $f(x^-)$ exist, in the last statement? i.e. ...