Questions tagged [piecewise-continuity]
For use when asking questions about piecewise continuous functions, their properties, or the functional branch itself.
507 questions
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Continuity of piecewise application between Metric Spaces
I have the following exercise and I don't know if my proof is correct:
Let $(X,d_{X}),(Y,d_{Y})$ metric spaces with $X=X_{1}\cup X_{2}$
Let $f_{i}:X_{i}\to Y, \ i\in\{1,2\}$ continuous applications ...
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Intervals of continuity of $ f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1}$
Let
$$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}\,dt.$$
Define
$$f(x)=\frac{\Gamma(x+1)}{\lfloor\Gamma(x+1)\rfloor+1},\qquad x\in[0,\infty).$$
Find the maximal open intervals $I_n=(a_n,b_n)\subset[0,\infty)...
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Is there a more efficient way to construct this 3D piecewise function?
I am building a piecewise function of two variables in 3D Cartesian space but, as currently written, it is utterly enormous. Is there a more compact way to implement it? (Context below!)
The Function
$...
- 2,153
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Show $\frac{B}{A}=-\frac{C_3}{C_2} \cdot \frac{3 \sqrt{3}}{4}$ for two pulse‐train Fourier coefficients
Question
Consider the two periodic signals
$$ y(t)=B\sum_{n\in\mathbb Z}\operatorname{rect}\!\bigl(2t-n\bigr),
> \qquad x(t)=A\sum_{n\in\mathbb
> Z}\operatorname{rect}\!\Bigl(\tfrac{t}{2}-6n\...
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What are the smoothness requirements for the curve on which a line integral is defined?
Wikipedia defines the line integral of a scalar field $\int_C f({\bf s})\, ds$ for a "piecewise smooth curve $C$". Unfortunately, there does not seem to be a consistent definition across the ...
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Can the derivative of a function always be Riemann integrated? [duplicate]
I realized I didn't know the answer to that question which seems simple, but I couldn't figure it out myself after reviewing my textbooks and draw a conclusion on my own.
Suppose we have a function $f:...
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Rourke and Sanderson Regular Neighbourhoods
Okay, I guess a regular neighbourhoods are supposed to be like a tubular neighbourhoods in smooth manifolds. (Am I correct?)
In Rourke and Sanderson's "Introduction to Piecewise Topology", I ...
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Full Subcomplexes - whole boundary?
I'm reading Rourke and Sanderson's "Introduction to Piecewise Linear Topology", and in Chapter 3, they have given a criteria for full subcomplexes. It goes like this (Page 31) :
Suppose $L \...
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Rourke and Sanderson Lemma 2.17 typo?
I'm reading Rourke and Sanderson's "Introduction to Piecewise Linear topology" book. I believe there is a typo in the proof of Lemma 2.17.
Its quite clear what the construction is (well, ...
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General position (Special Case) PL manifolds (Rourke & Sanderson)
I'm reading Rourke and Sanderson's Introduction to PL manifolds.
I was reading the proof of 5.3 on General position, and I do not understand the proof. (page 61)
First, let me go through a few ...
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Regarding Transversality (PL and Smooth)
I'm reading Rourke and Sanderson's "Introduction to PL manifolds".
In Chapter 5, the book gives a very complicated definition of "General position" and "Transversality", ...
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Product of PL manifolds is a PL manifold (Rourke & Sanderson)
I'm self studying Piecewise Linear Topology by Rourke and Sanderson's "An Introduction to Piecewise Linear Topology" book.
And I'm stuck on the following exercise : (Page 7, Exercise 1.9 (3))...
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How to integrate over a piecewise function?
I have an equation of the form
$$A=F(g(x))_{x}$$
with a given F being a piecewise function, where two parts are constant and don't depend on $g(x)$
$$\begin{eqnarray}
F(g(x)):=\left\lbrace\begin{array}...
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What is a piecewise smooth curve and what is not?
I just learned about Green's theorem and learned that it applies only to positively oriented, piecewise smooth and simple closed curves. However, I don't understand what are piecewise smooth curves ...
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Can I integrate this piecewise function? [closed]
Let $n > 2$ be an integer, and set $1 \le u \le n$. Let $f$ be a function with argument $x \in \mathbb{R}$, defined on $1 \le x \le n$ by the formula
$$f(x) := \begin{cases}
0 & x=\frac{n}{u} \...
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