Many have probably seen an example of a function that is continuous at only one point, for example
##f:\mathbb{R}\rightarrow\mathbb{R}\hspace{5pt}:\hspace{5pt}f(x)=\left\{\begin{array}{cc}x, & \hspace{6pt}when\hspace{3pt}x\in\mathbb{Q} \\ -x, &...
In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...
Now, there's this conventional definition of the Hölder continuity of a function ##f## defined on ##[a,b]\subset\mathbb{R}##:
For some real numbers ##C>0## and ##\alpha >0##, and any ##x,y\in [a,b]##, ##|f(x) - f(y)|<C|x-y|^{\alpha}##.
However, this does not include functions like ##f(x) =...
I know how Gauss law helps us to calculate the discontinuity at a point on the surface of a surface charge.
Similarly using Gauss law, is there a way to determine the continuity at other points of electric field due to a surface charge or the continuity at all points of electric field due to a...
Consider a magnetic dipole distribution in space having magnetization ##\mathbf{M}##. The potential at any point is given by:
##\displaystyle\psi=\dfrac{\mu_0}{4 \pi} \int_{V'} \dfrac{ \rho}{|\mathbf{r}-\mathbf{r'}|} dV' + \dfrac{\mu_0}{4 \pi} \oint_{S'}...
This is not so much a "Homework" question I am just giving an example to ask about a specific topic.
Homework Statement
Is ##f(t,y)=e^{-t}y## Lipschitz continuous in ##y##
Homework Equations
I don't really know what to put here. Here is the definitions...
Why can't G and its derivative be continuous in the relation below?
$$p(x)\dfrac{dG}{dx} \Big|_{t-\epsilon}^{t+\epsilon} +\int_{t-\epsilon}^{t+\epsilon} q(x) \;G(x,t) dx = 1$$
Can quantum cellular automata/quantum game of life simulate quantum continuous processes in the continuous limit?
At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document
it is said that: "For example, several works simulate quantum field theoretical equations in the...
Homework Statement
Let ##p\in\Bbb{R}##. Then the function ##f:(0,\infty)\rightarrow \Bbb{R}## defined by ##f(x):=x^p##. Then ##f## is continuous.
I need someone to check what I've done so far and I really need help finishing the last part. I am clueless as to how to show continuity for...
Hello everyone!
I'm a student of electrical engineering, preparing for the theoretical exam in math which will cover stuff like differential geometry, multiple integrals, vector analysis, complex analysis and so on. So the other day I was browsing through the required knowledge sheet our...
Homework Statement
[/B]
We have a function f(x) = |cos(x)|.
It's written that it is piecewise continuous in its domain.
I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be..
Homework Equations
[/B]
We say that a function f is...
Homework Statement
Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I cant find notes on it.It think it has something to do with limits. Many thanks.
Homework Statement
f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ.
Homework Equations
lim{x->a}f(x) = f(a)
The Attempt at a Solution
I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)
Homework Statement
Prove the continuity from below theorem.
Homework Equations
The Attempt at a Solution
So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity)...
1. The problem statement:
Let ##f:[a, b] \rightarrow \mathbb{R}##. Prove that if ##f## is continuous, then ##f## is bounded.
2. Relevant Information
This is the previous exercise.
I have already proved this result, and the book states to use it to prove the next exercise. It also hints to use...
Something about the theory of quantum measurement/collapse in the case of quantum fields... Suppose I have a field, either a scalar, vector or spinor, that I want to describe as a quantum object. For simplicity let's say that it's a scalar field ##\phi (\mathbf{x})##, where the ##\mathbf{x}## is...
Homework Statement
Let γs : I → Rn, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ0. Show that there exists some 0 < δ ≤ ε such that γs is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of...
Homework Statement
This is one test question we had today and it asks as to provide examples of functions and intervals. Some may be untrue so we had to identify it. The test isnt graded yet so these are my question answers. Hopefully you'll correct me where necessary and provide a true...
What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...
Homework Statement
f(x)= (sincx/x) ; x<0
1+(c)(tan2x/x) ; x≥0
Homework Equations
The Attempt at a Solution
Lim as x tends to 0+[/B] = 1+c⋅2⋅(sin2x/2x)⋅(1/cos2x) =1+c⋅2⋅1⋅1= 1+2c
Lim as x tends to 0 - = (sincx/x)=(c/1)⋅(sinx/x)=c⋅1=c
Equating both: 1+2c=c...
Homework Statement
Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##.
Homework Equations
##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##
The Attempt at a Solution...
A function f: R->R is a continuous function such that f(q) = q for every rational number q.
Prove f(y) = y for every real number y.
I know every irrational number is the limit to a sequence of rational numbers. But I not sure how to prove f(y) = y for every real number y. Any ideas?
Hi, this looks stupid and simple, but I just can't get my head around it.
Assuming a homogeneous medium.
The electromagnetic continuity equation goes as
∇⋅J + ∂ρ/∂t = 0
since J = σE, ρ = ɛ∇⋅E, and assuming the time dependence exp(-iωt)
we have
σ∇⋅E - iωɛ∇⋅E = 0
(σ - iωɛ)∇⋅E = 0
So, σ - iωɛ = 0...
Hello again,
Am facing a difficulty, the question is that ,
Is energy and momentum conserved for a particle in an infinite square well, at the boundary i.e, at x=a, where the potential suffers an infinite discontinuity??
V=0 for -a<=x<=a
V=infinite else-where
Thanks in...
So proofs are a weak point of mine.
The hint is that a composite of a continuous function is continuous. I'm not really sure how to use that. What I was thinking was something to the effect of an epsilon delta proof, is that applicable?
Something to the effect of:
##A \sim B\text{ and let } f...
Hello,
1. Homework Statement
1) Let f(x) continuous for all x and (f(x)2)=1 for all x. Prove that f(x)=1 for all x or f(x)=-1 for all x.
2) Give an example of a function f(x) s.t. (f(x)2)=1 for all x and it has both positive and negative values. Does it contradict (1) ?
2. The attempt at a...
Homework Statement
Homework Equations
lim(x,y)->(a,b)f(x,y) continuous at (a,b) if lim(x,y)->(a,b)f(x,y)=f(a,b)
Squeeze theorem if lim a=lim c and lim a<= lim b <= lim c then lim b= lim c
The Attempt at a Solution
I proved that all the limits exist but somewhat the functions aren't all...
Homework Statement
In a jet engine, a flow of air at 1000 K, 200 kPa, and 40 m/s enters a nozzle, where the air exits at 500 m/s and 90 kPa. What is the exit temperature, inlet area, and exit area, assuming no heat loss?
Homework Equations
min = mout = m
where m = mass air flow
dE/dt cv = Qcv...
This is picture taken from my textbook.
I understood the last two statements "To check whether..". A function is one if its strictly increasing or decreasing. But I am not able to understand the first statement. Polynomials are continuous functions. Also, a continuous function ± discontinuous...
Homework Statement
##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##:
The...