Continuous Definition and 1000 Threads

Let f be a continuous function defined on (a, b). Supposed f(x)=0 for all rational numbers x in (a, b). Prove that f(x)=0 on (a, b). i don't even know where to start...any tips just to point me in the right direction?

Hi, May be a dumb question; imagine a hypothetical situation of a spaceship in space with no influence of gravity due to Earth or nearby moon. Assuming the spaceship has enough fuel, if it injects the fuels outwards, it will accelerate in the opposite direction. Now, the new velocity will be...

Let n ≥ 2 be a natural number. Show there is no continuous function q_n : ℂ → ℂ such that (q_n(z))^n = z for all z ∈ ℂ. The only value of this function we can deduce is q_n(0)=0. Moreover any branch cut we take in our complex plane will touch zero. These two facts would make me a bit...

Prove that the inverse of a continuous injective function f:A -> ℂ on a compact domain A ⊂ ℂ is also continuous. So basically because we're in ℂ, A is closed and bounded, and since f is continuous, the range of f is also bounded. Given a z ∈ A, I can pick some arbitrary δ>0 and because f is...

Give me a function that is piecewise continuous but not piecewise smooth

What is an example of a continuous function f:\mathbb{R}\to\mathbb{R} such that f(\mathbb{R}) is open?

Homework Statement Using the definition of continuity, prove that the function f(x) = sin x is continuous. Hint: sin a − sin b = 2 sin (a-b)/2 . cos (a+b)/2Homework EquationsThe Attempt at a Solution Using the idea that: |sin(x)| ≤ |x| |cos(x)| ≤ 1 along with the hint: sin a − sin b = 2...

Homework Statement A function f:(a,b)\to R is said to be uniformly differentiable iff f is differentiable on (a,b) and for each \epsilon > 0, there is a \delta > 0 such that 0 < |x - y| < \delta and x,y \in (a,b) imply that \left|\frac{f(x) - f(y)}{x - y}-f'(x)\right| < \epsilon. Prove that...

Hi, I was searching the forum about comparing continuous probability distributions and came across this post back in 2005. "You could make two variables X(t) = value of the "true" disrtibution (expensive simulation) at point t and Y(t) = value of the alternative dist. (practical simulation)...

Hi everyone, I want to understand how these concepts work. Suppose that we have a signal x(t) which has a maximum frequency component of 3 Hz. So let the DTFT of this signal be like that: http://img341.imageshack.us/img341/1134/31096081.png Also let y[n] be the digital signal that we get...

What is an example of an absolutely continuous function on [a,b] whose derivative is unbounded? I know that the function f: [-1,1] defined by f(x) = x^2sin(1/x^2) for x ≠ 0, f(0) = 0 is continuous and its derivative f'(x) = 2xsin(1/x^2)-2/xcos(1/x^2) for x ≠ 0, f'(0) = 0 is unbounded on...

Homework Statement Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero. Homework Equations A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')}...

Homework Statement C[a,b] is a vector space of continuos real valued functions. for f,g in C[a,b] <f,g>=∫f(x)g(x)dx, [a,b] Give a completely rigorous proof that if <f,f>=0, then f=0 2. The attempt at a solution I tried to prove this by contrapositive, "f≠0 implies that <f,f>≠0 When...

Homework Statement for discrete basis vectors {{e_n}}, a state vector |psi> is represented by a column vector, with elements being psi_n = <e_n|psi>. When basis vectors correspond to those with continuous eigenvalues, vectors are represented by functions. Give such an example of a state...

Homework Statement Suppose that f(x) is a continuous function on [0,2] with f(0) = f(2). Show that there is a value of x in [0,1] such that f(x) = f(x+1). Homework Equations Intermediate Value Theorem? Extreme Value Theorem? Periodicity? The Attempt at a Solution For sure there's an...

hi i was wondering why the spectrum of an ordinary light bulb is continuous. i know that "i think it's thermodynamics" says that some temperature creates a specific continuous radiation, but how is this reconcilable with quantum mechanics and e.g. a sodium gas, that emits only a tiny yellow...

Hi! I have used the physics forum a lot of times to deal with several tasks that I had and now its the time to introduce my own query! So please bear with me :-) Homework Statement Equip the set C^1_{[0,1]} with the inner product: \left\langle f,g \right\rangle= \int_{0}^{1}...

Homework Statement Let f : R → R be continuous on R and assume that P = {x ∈ R : f(x) > 0} is non-empty. Prove that for any x0 ∈ P there exists a neighborhood Vδ(x0) ⊆ P. Homework Equations The Attempt at a Solution If you choose some x, y ∈ P, since f(x) is continuous then |f(x)...

I did the work. I'm not sure on some of these. I think for (c) I need to make D = (0, infinity) http://i111.photobucket.com/albums/n149/camarolt4z28/1-3.png http://i111.photobucket.com/albums/n149/camarolt4z28/2-3.png http://i111.photobucket.com/albums/n149/camarolt4z28/3-1.png

*couldn't edit the title so Find* Homework Statement f(x) = x+1 , x<0 f(x) = 1 , x=0 f(x)= x2-2x+1 , x>0 The Attempt at a Solution for a (find f'(0), if it exists) i did as followed Lim h->0- (x+h)+1-(x) /h giving me in the end 1 as for the third equation, I did...

let g:R->R be a real function defined by rule g(x) = x^2 if x\in\mathbb{Q} and g(x) = 0 if x\notin\mathbb{Q} is g continuous (*on R)? Many thanks in advance *thanks for pointing out mistake above.

Homework Statement The problem is: Suppose f is a function with the property f(x+y)=f(x)+f(y) for x,y in the reals. suppose f is continuous at 0. show f is continuous everywhere. I saw this post as an alternative solution that doesn't use epsilon-delta...

Homework Statement Let M, N be two metric spaces. For f: M --> N, define the function on M, graph(f) = {(x,f(x)) \inMxN: x\inM} show f continuous => graph(f) is closed in MxN Homework Equations The Attempt at a Solution I can't figure out what method to use. I have...

Homework Statement Suppose that the function f|(a,b)→ℝ is uniformly continuous. Prove that f|(a,b)→ℝ is bounded. Homework Equations A function f|D→ℝ is uniformly continuous provided that whenever {un} and {vn} are sequences in D such that lim (n→∞) [un-vn] = 0, then lim (n→∞) [f(un) -...

PROVE mean (X bar) of a continuous distribution is given by: [SIZE="7"] ∫x.f(x)dx {'a' is the lower limit of integration and 'b' is the upper limit}

Suppose a function f : R → R satisfy f(x + y) = f(x) + f(y) and f is continuous at x = 0: Show that f is continuous at every point in R. (Hint: Using the fact that lim f(x) = l implies x→x0 limf(x0+h)= l h→0 )

For the density function for random variable Y: f(y) = cy^2 for 0<= y <= 2; 0 elsewhere We are asked to find the value of c. I did a definite integral from 0 to 2 of cy^2. I get c = 3/8. Why would the book show an answer of c = 1/8? Is this an error on their part or am I missing something...

say we have a sequence of probability measures on R, such that each one is abs. continuous wrt the Lebesgue measure... is it possible that these measures converge weakly to a measure which is _not_ abs. continuous wrt the Lebesgue measure?

I suppose my question is, "does the set of all continuous functions comprise a continuum?" How would one even start at trying to prove that? Any ideas or suggestions?

[SIZE="6"]Name 5 signals and the systems that process them. – Draw the block diagrams to show how the signal gets transformed. • Choose both Continuous and Discrete signal • Include some examples from Bio Medical Engineering.

Ok so mathematically you can divide any number by any other (nonzero) number and you can keep dividing that number however many times you want. Like dividing 1 by 2 and then by 2 again etc. And this is the basis of the famous paradox that mathematically, you can't really move from point a to b...

Can you give some examples of the infinite sets that are uncountable and that are not continuous? I know the infinite sets that are countable and discrete, and I know the continuous sets, but couldn't find an example for the above situation.

hi i was thinking that after the laws of quantum mechanics, where all atoms and molecules have discrete energy states, things like continuous spectrums should be forbidden. but why are there still molecules like chlorophyl which show this property? i would think, that this means, that the...

I am teaching myself quantum mechanics and have just read the particle in a box explanation, which is the first derivation of a theoretical reason why only discrete energy levels are possible within certain bound scenarios. In Shankar, the argument uses a requirement that the wave function...

Homework Statement Discrete-time transforms What does it mean when it says this : These transforms have a continuous frequency domain: Discrete-time Fourier transform Z-transform What is the meaning of continuous frequency domain..?

Continuous absorption spectrum -- why this happens? Homework Statement A pure green glass plate placed in the path of light, absorbs everything everything except green, similarly red glass plate absorbs everything except red. Homework Equations May i know the reason for this? Thanks...

Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Giv Homework Statement Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Give the derivation The Attempt at a Solution E(|X - b|) E[e - \bar{x}] = E(X) E(|E[e - \bar{x}] - b|)...

Homework Statement While dealing with continuous absorption spectrum, my book depicts like this " A pure green glass plate when placed in the path of white light, absorbs everything except green and gives continuous absorption spectrum" Homework Equations The Attempt at a Solution...

Continuous Bijection f:X-->X not a Homeo. Hi, All: A standard example of a continuous bijection that is not a homeomorphism is the map f:[0,1)-->S^1 : x-->(cosx,sinx) ; for one, S^1 is compact, but [0,1) is not,so they cannot be homeomorphic to each other. Now, I wonder...

So I've been reading about orientated surfaces lately, and I always see the definition that a surface S is orientable if it is possible to choose a unit normal vector n, at every point of the surface so that n varies continuously over S. However, what does "varies continuously" mean? I never...

I'm writing a little program for generating some images, and at one point I need to calculate how much of a circle is on either side of a straight line that bisects the circle. The line is always vertical so it is easy to get the value of how much of a horizontal line segment within the circle...

Continuous, Onto Function: (0,1)x(0,1)-->R^2 Hi, All: Just curious about finding a continuous onto function from the open unit square (0,1)x(0,1) into R^2. All I can think is that the function must go to infinity towards the edges, because if it could be continued into the whole square...

Can someone tell me how to obtain the figure in red circle? I've trying out on my own but can't get that.

I understand the classical view of EM fields as being (theoretically) continuous. What I don't quite get is how this can be reconciled with the QM view of photons coming only in fixed frequencies (The electromagnetic field may be thought of in a more 'coarse' way.). Is the number of possible...

I've been reading Ballentine, Chapter 1. Have I got this the right way around? Taking our inner product to be linear in its second argument and conjugate linear in its first, the (continuous?) conjugate space of a Hilbert space \cal{H} is the following set of linear functionals, each identified...

Or do we have no idea?

Prove that the function is continuous when f(x)=0 f(x)=x4-7x3+11x2+7x-12f(c)-\epsilon<f(x)<f(c)+\epsilon Limits maybe taken, however, we do not have the value for c in the limit equation.

Homework Statement Let F(x)=\begin{cases} .25e^{x} & -\infty<x<0\\ .5 & 0\leq x\leq1\\ 1-e^{-x} & 1<x<\infty\end{cases}$. Find a CDF of discrete type, F_d(x) and of continuous type, F_c(x) and a number 0<a<1 such that F(x)=aF_d(x)+(1-a)F_c(x) Homework Equations The Attempt at a...

Homework Statement Define the function at a so as to make it continuous at a. f(x)=\frac{4-x}{2-\sqrt{x}}; a = 4 Homework Equations \lim_{x \rightarrow 4} \frac{4-x}{2-\sqrt{x}} The Attempt at a Solution I cannot think of how to manipulate the denominator to achieve f(4), so I...

Hi all, Could anyone guide me on the following prove √2 = 1+1/(2 + 1/(2+ 1/(2+ 1/(2+···))))