Continuous Definition and 1000 Threads

Suppose that $f(\theta)$ is a continuous periodic piecewise differentiable function. Prove that $f(\theta) = f(0) + \int_0^{\theta}g(t)dt$ for a piecewise continuous $g$. I just need a nudge in the right direction here.

Homework Statement Fix a real number a>1. If r=p/q is a rational number, we define a^r to be a^(p/q). Assume the fact that f(r)=a^r is a continuous increasing function on the domain Q of rational numbers r. Let s be a real number. Prove that lim r--->s f(r) exists Homework Equations...

Homework Statement Find the expected value of a continuous variable y with pdf fy= alpha*y^-2, 0<y<infinity. I know it is the integral from zero to infinity of y*fy, but I don't know where to go from there. I'm then supposed to use the expected value to find the method of moments...

In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions? I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions...

I am given a continuous charge problem in which there is a non-conducting wire of legnth L lying along the y-axis and I am required to calculate the electric field at any point along the x-axis. I know how to compute the electric field of a continuous charge distribution at a given point, but...

If f_n : A\rightarrow R sequnce of continuous functions converges uniformly to f prove that f is continuous My work Given \epsilon > 0 fix c\in A want f is continuous at c |f(x) - f(c) | = |f(x) - f_n(x) + f_n(x) - f(c) | \leq |f(x) - f_n(x) | + |f_n(x) - f(c) | the first absolute...

Homework Statement Suppose that f is an odd function satisfying \mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0). Prove that f(0)=0 and f is continuous at x=0. Homework Equations The Attempt at a Solution Since f is an odd function f(0) = - f(0) \Rightarrow f(0) = 0 Let...

Hey guys, I have been working on the following question: http://imageshack.us/a/img407/4890/81345604.jpg For part a f and g are continuous on I => there exists e > 0 and t_0 s.t. 0<|{f(t) - g(t)} - {f(t_0) - g(t_0)}| < e using |a-b| >= |a| - |b|, |{f(t) - g(t)} -...

Homework Statement Find a value for the constant k that will make the function below continuous: f(x)=\frac{x-1}{x^2-1}\ \text{if}\ x<=0 f(x)=\frac{tankx}{2x}~\text{if}~x>0 Homework Equations The Attempt at a Solution I've tried the only solution I can think of, which is to...

Homework Statement Show that the mapping f carrying each point (x_{1},x_{2},...,x_{n+1}) of E^{n+1}-0 onto the point (\frac{x_{1}}{|x|^{2}},...,\frac{x_{n+1}}{|x|^{2}}) is continuous. [b]2. Continuity theorems I am given. A transformation f:S->T is continuous provided that if p is a limit...

Hi all, I need help with a paragraph of my book that I don't understand. It says: "the map sending all of ℝ^n into a single point of ℝ^m is an example showing that a continuous map need not send open sets into open sets". My confusion arising because I can't figure out how this map can be...

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 Suppose h(t) is a causal signal and has the even part h_e(t) given by: h_e(t)= t[u(t)-u(t-1)]+u(t-1) for t>0. Find h(t) for all tHomework Equations For an even function f(x), f(x) = f(-x) Also even functions can be expressed as x_e(t) = 1/2[x(t)+x(-t)]The Attempt at a...

Homework Statement Let f be an increasing function defined on an open interval I and let c ϵ I. Suppose f is continuous at c. Prove sup{f(x)|x ϵ I and x < c} = f(c) Homework Equations The Attempt at a Solution Since I is an open interval and c is not able to be an end point...

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 For my proof, it tells me that f:X->Y is bijective. I understand that it is one-to-one and onto, but I just want to be clear about this from a neighborhood (open subset by our def) standpoint. Just to be clear: if f is bijective continuous, then that means the for all open...

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...

What does it mean to be "strictly-strictly" continuous? I am unsure what it means to be "strictly-strictly" continuous. Is that the same thing as saying just "strictly" continuous? Here is the context: \alpha is a unital *-homomorphism from M(A) to \mathcal{L}(A) such that \alpha is...

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?

Dear Sir… I am looking for a discrete counter part of a continuous variable. the continuous version of energy term in a liquid crystal is given by [\vec{n}\cdot(\nabla\times\vec{n})]^2. This is a square of a dot product between a vector 'n' and its curl field. My question is what is the exact...

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...