Continuous Definition and 1000 Threads

Showing the sum of functions are uniformly continuous Homework Statement Suppose f and g are uniformly continuous on an interval I. Prove f + g are uniformly continuous on I. Homework Equations The Attempt at a Solution Let ε >0 By definition, since f and g are uniformly...

Where is the fallacy in this "proof" that the Fourier series of f(x) converges to f(x) if f is continuous at x and has period 2π? (I read in Wikipedia that a counterexample had been provided). Start with the Dirichlet integral for the N-th partial sum of the (trigonometric) Fourier series...

I am trying to understand the theorem: Let f:S->T be a transformation of the space S into the space T. A necessary and sufficient condition that f be continuous is that if O is any open subset of T, then its inverse image f^{-1}(O) is open in S. First off, I don't really understand what...

Homework Statement Find values for a and b that ensure f is a continuous function if f(x) = ax + 2b if x ≤ 0 x2 +3a - b if 0 < x ≤ 1 2x - 5 if x > 1 Homework Equations The Attempt at a Solution ax + 2b = 2x...

Homework Statement Graph the function defined by the following. B = {(r/r0)B0 for r ≤ r0 {(r0/r)B0 for r > r0 (a) Is B continuous at r = r0? yes no (b) Is B differentiable at r = r0? Homework Equations The Attempt at a Solution I'm not exactly sure what to do...

Homework Statement I must calculate the characteristic function as well as the first moments and cumulants of the continuous random variable f_X (x)=\frac{1}{\pi } \frac{c}{x^2+c^2} which is basically a kind of Lorentzian.Homework Equations The characteristic function is simply a Fourier...

Homework Statement { 2x if x<1 h(x)= { cx^2+d if 1<=x<=2 { 4x if x>2 Homework Equations The Attempt at a Solution It tried taking the limit of 2x and cx^2+d at x->1 (from both sides) and set them equal to each other. I did the...

Homework Statement Use the definition of continuity to prove that the function f defined by f(x)=x^(1/2) is continuous at every nonnative number. Homework Equations Continuity in this text is defined as Let I be an interval, let f:I→ℝ, and let c be in I. The function f is continuous...

Homework Statement Find k so that the following function is continuous on any interval. j(x) = {k cos(x), x ≤ 0 {10ex − k, 0 < x Homework Equations The Attempt at a Solution I originally thought i had to check if the limits of both parts of the functions...

Homework Statement Determine all continuous functions g: R -> R such that g(x + y) = g(x) + g(y) for all x, y \in \mathbf{R} The Attempt at a Solution g(x) = g(x + 0) = g(x) + g(0). Hence G(0) = 0. G(0) = g(x + -x) = g(x) + g(-x) = 0. Therefore g(x) = -g(-x). It seems obvious that the only...

Homework Statement It says: \displaystyle f:{{\mathbb{R}}^{2}}\to \mathbb{R} \displaystyle f\left( x,y \right)=\left\{ \begin{align} & 1\text{ if 0<y<}{{\text{x}}^{2}} \\ & 0\text{ in other cases} \\ \end{align} \right. Show that all the directional derivatives about (0,0) exist but f...

Homework Statement Let \displaystyle f:{{\mathbb{R}}^{n}}\to \mathbb{R} a continuous function. Proove that: If \displaystyle f\left( p \right)>0 then there's a ball \displaystyle {{B}_{p}} centered at p such that \displaystyle \forall x\in {{B}_{p}} we have \displaystyle f\left( x...

Homework Statement Prove f(x)=\sqrt{x^{2}+1} is uniformly continuous on the real line. Homework Equations Lipschitz Condition: If there is a constant M such that |f(p) - f(q)| \leq M |p-q| for all p,q \in D, then f obeys the Lipschitz condition. Mean Value Theorem: Let f be continuous on...

"Absolutely continuous r.v." vs. "continuous r.v." I've recently come across the term "absolutely continuous random variable" in a book on measure theoretic probability. How am I supposed to distinguish between AC random variables and just continuous random variables?

Hi, I'm beginning to learn QM, and I've never seen any treatment of vector spaces with infinite bases. Countable case is quite digestible, but uncountable just flies over my head. Can anyone recommend me place where to learn this more advanced part of linear algebra, with focus on stuff...

I'm self-studying Griffith's Intro to Quantum Mechanics, and on page 100 he makes the claim that the eigenfunctions of operators with continuous spectra are not normalizable. I can't see why this is necessarily true. Hopefully I am not missing something basic. Thanks in advance.

Homework Statement Suppose that f is continuous and that there exist constants A,B ≥ 0 and k>1 such that |f(z)|≤A|z|−k for all z such that |z|>B. let CR denote the semicircle given by |z| = R, Re(z) ≥ 0. Prove that limR→∞∫f(z)dz=0 Homework Equations The Attempt at a Solution I...

The last days I have been thinking about the following question. How does standard QM explain the continuous spectrum in beta-decay? Why can the created electrons (and, hence, also the neutrinos) in beta-decay acquire any possible energy within a certain range as long as their sum conserves...

Here's an interesting question--I've asked some faculty members around here and "off the top of their head" none of them knows the answer. My gut says "yes", but my gut sucks at math. So here's the statement: Suppose we have a function f:\mathbb{R}^2\to\mathbb{R}, with the property that for...

Homework Statement For all real numbers, f is a function satisfying |f(x)|<=|x|. Show that f is continuous at 0 Homework Equations The Attempt at a Solution Really stuck on this cause I'm confused with the absolute values on this function. I *think* to show this you have to...

Homework Statement f(x) = {2x2 + x +3, x < 0 \frac{3}{x + 1} x ≥ 0 The 2 should be wrapped as 1 with a { but do not know how to do that. Homework Equations The Attempt at a Solution I was wondering if the squeeze rule would be...

Let X be a complex Banach space and T in L(X,X) a linear operator. Assuming only that (T*f)(x)=f(Tx), where x in X and f in X* how can I prove that T is continuous?

Homework Statement Find the continuous branch cut of a complex logarythm for C\[iy:y=>0] One of the complex numbers, for example, is -4i Homework Equations I don´t understand what to do with the subset. How could I find the continuous branch cut in the subset? The Attempt at a...

We know that the \mathcal L\{f(t)\} = \int^{\infty}_0 e^{-st}f(t) dt. Say we want to, for example, solve the following IVP: y'' + y = f(t) where f(t) = \begin{cases} 0 & 0 \leq t < \pi \\ 1 & \pi \leq t < 2\pi\\ 0 & 2\pi \leq t \end{cases} and y(0) = 0 , y'(0) = 0 We apply Laplace on both...

The Integration by Parts Theorem states that if f' and g' are continuous, then ∫f'(x)g(x)dx = f(x)g(x) - ∫f(x)g'(x)dx. My question is, are those assumptions necessary? For example, this holds even if only one of the functions has a continuous derivative (say f' is not continuous but g'...

if f : (0, 1]--> R is given by f(x) = 0 if x is irrational, and f(x) = 1/(m+n) if x = m/n in (0, 1] in lowest terms for integers m and n. How can i prove that this function is continuous at 1/√2?

Homework Statement I am trying to come up with a continuous function in L1[0,infinity) that doesn't converge to 0 as the function goes out to infinity. Homework Equations I am trying to show an example of an f in L1[0,infinity) (i.e. ∫abs(f) < infinity) where the limit as the function...

I am trying to solve a question and I need to justify a line in which |lim(x-->0)(f(x))|≤lim(x-->0)|f(x)| where f is a continuous function. Any help?

continuous -- how can I combine these open sets Homework Statement let ##X,Y## be compact spaces if ##f \in C(X \times Y)## and ## \epsilon > 0## then ## \exists g_1,\dots , g_n \in C(X) ## and ## h_1, \dots , h_n \in C(Y) ## such that ##|f(x,y)- \Sigma _{k=1}^n g_k(x)h_k(y)| < \epsilon...

Homework Statement Take f: (a,b) --> R , continuous for all x0in (a,b) and take (Ω = (a,b) , F = ( (a,b) \bigcap B(R)) where B(R) is the borel sigma algebra Then prove f is a borel function The Attempt at a Solution I know that continuity of f means that for all x in (a,b) and all...

Hi All. I may sound weird and I know I am wrong somewhere. But a little explanation would really help. A system with 1 degree of freedom(d.o.f) has 1 governing differential equation. Similarly a system with 2 d.o.f has 2 (coupled) differential equations and so on. But a continuous system has...

How do you find all the values of "a" such that f is continuous on all real numbers? Find all values of a such that f is continuous on \Re f(x)= x+1 if x\leq a x^2 if x>a I tried solving but i do not even know where to start! Please help!

A) Let us say that we have some arbitrary sequence of natural numbers. e.g. 1, 2, 7, 3, 17, 19. Is it possible to convert every finite and infinite sequence into some continuous function model, such as in Fourier theory? I know that it is possible to extract some discrete samples from a...

Homework Statement Determine the values of k,L,m and n such that the following function g(x) is continuous and differentiable at all points Homework Equations 2x2-n if x<-2 mx+L if -2≤x<2 kx2+1 if x≥2 The Attempt at a Solution So I know that...

I recall reading somewhere that the mean value of a continuous variable is situated at a point that acts as a fulcrum about which all other values are considered "weights". In other words, if we define the mean as μ = \int^{∞}_{-∞} x ρ(x) dx (where rho is the probability density) then...

Homework Statement In the winter, the monthly demand in tonnes, for solid fuel from a coal merchant may be modeled by the continuous random variable X with probability density function given by: f(x)=\frac{x}{30} 0≤x<6 f(x)=\frac{(12-x)^{2}}{180} 6≤x≤12 f(x)=0 otherwise (a)...

I was wondering what the majority opinion was on this issue, among physicists and philosophers as well. I can't imagine zooming in a million times smaller than the plank length and still not being at a smallest length, however a discrete universe doesn't make much sense to me. Are there any...

Hello, Sorry if the question sounds silly, but can a continuous matrix be seen as a differential operator? First of all, let me state that I have no idea what a continuous matrix would formally mean, but I would suppose there is such an abstract notion, somewhere? Secondly, let me tell...

Simple continuous tracking radar system with cantenna We are in need of a simple radar target tracking signal processing method. It's to be very short range, less than 5 meters. It needs to track a single target that can be moving or stationary at any time. We are using 2.4 GHz ISM band...

Homework Statement Let f : (-1, 1) → ℝ. f satisfies the intermediate value property and is one-to-one on (-1, 1). Prove f is continuous on (-1, 1) The Attempt at a Solution I was thinking that the IVP and one-to-one implies that f should be strictly monotonic and that a strictly monotonic...

Homework Statement Find the value of p and q that make the function continuous Homework Equations f(x)= x-2 if x≥2 \sqrt{p-x^{2}} -2<x<2 q-x if x≤-2 The Attempt at a Solution lim f(x)= x-2 n→2+ lim f(x)=q-x n→-2 I really have no idea how to continue,the teacher never explained this and I...

Suppose that f is continuous function on the interval [a,b] integral from b to a If(x)I dx =0 if and only if f(x)=0 for all x in [a,b] ıs it true or false ? ı can prove that if f is zero,integral is zero but ı can,'t do that if integral is zero f is zero Regards

Hi, can someone give me pointers on this question Homework Statement Prove or provide a counterexample: If f : E -> Y is continuous on a dense subset E of a metric space X, then there is a continuous function g: X -> Y such that g(z) = f(z) for all z element of E. The Attempt at a Solution...

Hi, can someone please check if my proof is correct 1. a) Assume f : R -> R is continuous when the usual topology on R is used in the domain and the discrete topology on R is used in the range. Show that f must be a constant function. My attempt : Let f: R --> R be continuous. Suppose...

What proof or evidence -both theoretical or emperical - is there, that spacetime is quantized or continuous?

Hi, I am correcting a C code which was written for a flame machine. My desire is to give a burst cool time when the machine working continuously. In deep, user could have the facility to enter the flame time and cool time before a continuous operation. So, i exactly want to off the...

Hi, I am trying find equations for continuous "stretchy" collisions, in other words, I have two perfectly round objects of known mass, radius, and velocity, and want to collide them and be able to have them squish together and then bounce apart. I am aware of the method of solving for the...

Hello Forum, there are two typeos of servo motors. One of them is called continuous. does it mean that it can rotate to any angular position, between 0 degrees and 360? The other type (the non continuous one) still rotates at any angular position, but only within a limited range, like 0 to...

Homework Statement T or F, If f:ℝ→ℝ is continuous on a dense set of points in ℝ, then f is continuous on ℝ. Homework Equations definition of continuity using sequences, maybe? The Attempt at a Solution false. Take f(x)= {1 if x\in Q(rational numbers) and 0 if x\in...

Homework Statement Show that if f: [a,b]→Re is increasing and the range of f is a bounded interval then f is continuous. Homework Equations N/A The Attempt at a Solution I have no idea where to start, but I decided to start with a couple of things. Proof: Let f: [a,b]→Re...