# Questions tagged [piecewise-continuity]

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

480 questions
Filter by
Sorted by
Tagged with
39 views

### 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 ...
• 185
69 views

• 121
80 views

• 341
88 views

### 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 ...
• 1,079
1 vote
31 views

### 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 ...
• 109
60 views

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,..... • 121 0 votes 1 answer 35 views ### 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 $$f(x)... 2 votes 1 answer 82 views ### Finding the derivative of a function with two absolute values within it, using a piecewise function I'm trying to solve some a problem relating to absolute values. I found online the strategy for solving similar functions from here: https://www.youtube.com/watch?v=eIHtq67nh7w&list=... 0 votes 0 answers 64 views ### How do I calculate the values for W, X, Y, and Z in this project, given specific constraints? I am exploring the math of smooth motion. I have everything sorted out for zero through four degrees of smoothness, but I'm stuck on some of the transition-time values for the fifth degree of ... • 1,759 1 vote 1 answer 36 views ### f(0) = 0, f(1) = 1 and f(n)=f(n-T_{m-1})-f(T_m-n) if T_m:=\frac{m(m+1)}2\ge n>T_{m-1}. Find the smallest n such that f(n) = 4. Here's the question: For nonnegative integer n, the following are true:$$ \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} - ... • 119 0 votes 0 answers 27 views ### 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 (... 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 : $$f^{(k)}(-1) \neq f^{(k)}(1) , k \in \{0,1,\cdots,n\}... • 25.9k 2 votes 2 answers 98 views ### finding the values of a and b(real numbers) that makes the function differentiable at any point of its domain I'm given this function:$$ 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 ... • 23 3 votes 1 answer 98 views ### 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{... • 33 5 votes 1 answer 383 views ### 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.... 0 votes 0 answers 26 views ### 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{... 0 votes 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 ... 0 votes 0 answers 22 views ### 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: [-... • 2,749 0 votes 0 answers 107 views ### 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 ... 0 votes 0 answers 24 views ### 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 ... 0 votes 0 answers 23 views ### 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 (... 0 votes 1 answer 35 views ### 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 ... • 401 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}$$with \alpha=... • 111 7 votes 1 answer 227 views ### Is there a good way to simplify this expression? The Short Version Is there a way to simplify this expression?$$ \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 × \... • 1,759 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 ... • 3,247 1 vote 1 answer 49 views ### 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,... • 1,390 1 vote 1 answer 42 views ### Are these two piecewise functions equivalent or different? I got this solution to the inequality$|x - 5| + |x - 4| \ge 3:$$$|x - 5| + |x - 4| - 3 = \begin{cases} 2x - 12& \text{if }x > 5\\ -2 & \text{if }4 \le x \le 5\\ -2x + 6& \text{if }x &... • 380 4 votes 2 answers 329 views ### Conflicting results for \int\frac{|x+1|}{x}\,\mathrm dx I've encountered an obstacle while trying to solve the following integral:$$\int{\frac{\sqrt{x^2+2x+1}}{x}dx}$$First thing we shall do is see that under the square root is actually (x+1)^2. When I ... • 2,144 2 votes 2 answers 123 views ### Piecewise limit does not exist proof [closed] Let$$f(x)=\begin{cases} 1&\text{if }x\leq 0\\ -1&\text{if }x>0 \end{cases}$$Prove, using the precise definition of the limit, that \lim\limits_{x\rightarrow 0} f(x) does not exist. I ... • 21 3 votes 1 answer 177 views ### Why would the |f(x)| be non differentiable for x belonging to Real Numbers? I'm just started calculus and come across a statement in 1 of my books (VG Advanced Problems in Mathematics) that goes like: If y = f(x) is differentiable for x belonging to the set of Real numbers, ... 0 votes 1 answer 306 views ### Discontinuities of Piecewise functions with undefined points In multiple class examples, and internet examples on discontinuities. I often see that the undefined points are often called "the points at which the function is discontinuous". So If I have ... • 348 0 votes 1 answer 29 views ### Indefinite as sumand My question is based on some questions about piecewise functions. The problem is when I "sum" two piecewise functions that don't share any interval. When I draw the graphic of this function ... 2 votes 0 answers 45 views ### Fourier Series Derivative of saw tooth-like function So I have this periodic signal x(t),$$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}$$and I was ... • 65 0 votes 0 answers 24 views ### Calculate {f^{\ast}}'(0) if f^{\ast} is an odd expansion of the function f which is defined on the interval 0\leq x\leq L Imagine that the function f(x) is defined on the interval 0\leq x\leq L. f is defined and continuous on [0,L] and differentiable on (0,L). also f(0)=f(L)=0 (The function f is the initial ... • 411 5 votes 1 answer 108 views ### Proving discontinuity at \,x =1\, using \,\varepsilon-\delta\, definition I was doing a proof for discontinuity using \,\varepsilon-\delta\, definition but I’m not sure whether the proof is right. Would you mind checking it for me please, thanks! g(x)=\begin{cases}\... • 87 0 votes 1 answer 121 views ### The number of disjoint intervals over which the function f(x) = |0.5x^2−| x | | is decreasing is The number of disjoint intervals over which the function f(x) = |0.5x^ 2−| x | | is decreasing is A)one B)two C)three D) none of these I actually solved it and the answer is three but I want to ... • 13 2 votes 1 answer 51 views ### Most efficient way to find the "solution intervals" / "case conditions"? (problem regarding a function with absolute value notation): I am trying to write the function f(x) = |x^2-1|+|x|-1 without the notation for the absolute value. "||" It makes logical sense to consider four cases, because each term in the absolute ... • 67 4 votes 1 answer 59 views ### Need help choosing delta to prove continuity for f(x)=\frac1x at x=\frac12 in a piecewise function. I am trying to prove continuity of the function$$\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)... • 479 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: f(x) = \begin{cases}\cos(x) & \text{ if } \frac{-\pi}{2}\leq x \leq \frac{\pi}{2}\\ 0 &\text{ otherwise.}\end{cases}\end{...
• 107
1 vote