Continuity Definition and 876 Threads

Let a\in\mathbb{R}, a>0 be fixed. We define a mapping \mathbb{Q}\to\mathbb{R},\quad q\mapsto a^q by setting a^q=\sqrt[m]{a^n}, where q=\frac{n}{m}. How do you prove that the mapping is locally uniformly continuous? Considering that we already know what q\mapsto a^q looks like, we can define...

Hello everyone. Last week I had an exam in advanced calculus. One of the questions asked about the continuity of a function of three variables. However, the doctor gave me 0 out of 3 for the question while i am sure that my answer is correct and i told him that but he insisted that its wrong...

First off, it's: $$x = 1+x^3$$ Turned into function as: $$f(x) = x^3 - x + 1$$ From my understanding, we need to find an interval in which x will be one more than it's cube. Giving some points, I started off with (0,1), (1,1), (-1,1), and (-2, -5). Where I'm confused is how and where do I...

I just started Calculus 1, a summer quarter that's compressed and I'm having trouble understanding a theorem that state continuity of the inverse function. Within my textbook, it mentions "If f(x) is continuous on an interval I with range R, and if inverse f(x) exists, then the inverse f(x) is...

Hello, i had studied the problem in 1D, but i thinking the problem in 2d, an i have the following question: in a potential -V between (-a,a) an 0 otherwise. One dimensional case: One of the boundary condition are : ##\phi_I \in (-a,a)##, and ##\phi_{II} \in (a,\infty)## ##...

Let $f:[0,\infty)\to\mathbb R$ be a differentiable function such that for all $a>0$ exists a constant $M_a$ such that $|f'(t)|\le M_a$ for all $t\in[0,a]$ and $f(t)\xrightarrow[n\to\infty]{}0.$ Show that $f$ is uniformly continuous. Basically, I need to prove that $f$ is uniformly continuous...

Hello! (Wasntme) I am looking at the following exercise: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a continuous function with the following property: $\forall \epsilon >0 \exists M=M( \epsilon)>0 \text{ such that if } |x| \geq M \text{ then } |f(x)|< \epsilon$. Show that $ f$ is uniformly...

Homework Statement Steam at 2 MPa and 208°C enters a nozzle with 20m/s. During the expansion process, its enthalpy drops to 2.86 MJ/kg because of the losses encountered. a) Determine the exit velocity from the nozzle. b) If the mass flow rate is 1kg/s, determine the flow area at the...

I understand the reasoning behind the equations ∫SJ.dS=-dQ/dt and thus ∇.J=-∂ρ/∂t. where the integral is taken over the closed surface S. However I'm a little confused about the conditions of steady currents: The book I'm using sets dQ/dt=0 and ∂ρ/∂t=0 in these cases. I don't understand this...

Let $A_{n\times n}$ be a matrix with real and distincts eigenvalues. Let $u(t,x_0)$ be a solution for the initial value problem $\overset{\cdot }{\mathop{x}}\,=Ax$ with $x(0)=x_0,$ then show that for each fixed $t\in\mathbb R,$ we have $$\lim_{y_0\to x_0}u(t,y_0)=u(t,x_0).$$

Example 1 in James Munkres' book, Topology (2nd Edition) reads as follows: Munkres states that the map p is 'readily seen' to be surjective, continuous and closed. My problem is with showing (rigorously) that it is indeed true that the map p is continuous and closed. Regarding the...

If we consider a perfect relativistic fluid it has energy momentum tensor $$T^{\mu \nu} = (\rho + p) U^\mu U^\nu + p\eta^{\mu \nu} $$ where ##U^\mu## is the four-velocity field of the fluid. ##\partial_\mu T^{\mu \nu} = 0## then implies the relativistic continuity equation...

If $$f(x,y)$$ be a continuous function of $$(x,y)$$ in the rectangle $$R:{a \leq x \leq b, c \leq y \leq d}$$ , then $$\int_a^b f(x,y) dx$$ is also a continuous function of $$y$$ in $$[c,d]$$ How to proceed with the proof of the above theorem?

Hi Lets say that f(x) is continuous. Then \int_0^x \! f(t)dt=G(x) is continuous. (I don't think you have to say that f need to be continuous for this, all we need to say is that f is integrable?, or do we need continuity of f here?) But my main question is about the converse. let's say...

A uniformly convergent sequence of continuous functions converges to a continuous function. I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I...

Homework Statement Theorem: Let f:[a, ∞)→ R. The following are equivalent. i) lim ƒ(x) = A as x→∞ ii) For all sequences {xn in [a,∞) with lim xn = ∞ we have lim f(xn) = A. Homework Equations For any ε > 0, |ƒ(x)-A| < ε if x < N The Attempt at a Solution I probably have this wrong, but I...

Suppose I have a function f(x) \in C_0^\infty(\mathbb R), the real-valued, infinitely differentiable functions with compact support. Here are a few questions: (1) The function f is trivially uniformly continuous on its support, but is it necessarily uniformly continuous on \mathbb R? (2) I...

If g(a) \neq 0 and both f and g are continuous at a, then we know the quotient function f/g is continuous at a. Now, suppose we have a linear operator A(t) on a Hilbert space such that the function \phi(t) = \| A(t) \|, \phi: \mathbb R \to [0,\infty), is continuous at a. Do we then know that...

Homework Statement Prove that \sqrt{x} is continuous in R+ by using the epsilon-delta definition. Homework Equations A function f from R to R is continuous at a point a \in R if : Given ε> 0 there exists δ > 0 such that if |a - x| < δ then |f(a) - f(x)| < ε The Attempt at a...

There is an apparent conflict between relativity and quantum theory, in which case quantum theory must be redundant isn't it as it explicitly makes the assumption that spacetime is continuous whereas relativity in-fact derives the notion that spacetime is continuous from an experimentally...

Homework Statement Hi I can't follow the derivaton in this link. It is the following equality they have in the beginning, which I don't understand: \nabla \cdot u = \frac{1}{\rho}\frac{d\rho}{dt} Following the very first equation on the page, I believe it should be \nabla \cdot u =...

Homework Statement Discuss the continuity and differentiability of f(x) = \begin{cases} x^2 & \text{if } x\in \mathbb{Q} \\ x^4 & \text{if } x\in \mathbb{R}\setminus \mathbb{Q} \end{cases} Homework Equations The Attempt at a Solution From the graph of ##f##, I can see...

1) For the following choice of A, construct a function f: R → R that has discontinuities at every point x in A and is continuous on the complement of A. A = { x : 0 < x < 1} My function is f(x) = 10 if x in (0,1) and Q and f(x) = 20, if x in (0,1) and irrational number, f(x) = 30, elsewhere...

Given [a,b] a bounded interval, and f \in L^{p} ([a,b]) 1 < p < \infty, we define: F(x) = \displaystyle \int_{a}^{x} f(t) dt, x \in [a,b] Prove that exists K \in R such that for every partition: a_{0} = x_{0} < x_{1} < ... < x_{n} = b : \displaystyle \sum_{i=0}^{n-1} \frac{| F(x_{i+1}) -...

Want to show that f(x)/g(x) is continuous as x goes to c given that g(c) is not 0 and f(c) exists. |f(x)/g(x) - f(c)/g(c)| = |1/(g(x)g(c))||f(x)g(c)-f(c)g(x)| = |1/(g(x)g(c))||f(x)g(c)-f(c)g(x)-f(x)g(x) + f(x)g(x)| <= |1/(g(x)g(c))|||f(x)||g(x)-g(c)| + |g(x)||f(x)- f(c)|| Now I am stuck

Homework Statement Prove whether f(x) = x^3 is uniformly continuous on [-1,2) Homework Equations The Attempt at a Solution I used Lipschitz continuity. f has a bounded derivative on that interval, thus it implies f is uniformly continuous on that interval. But as it is not a...

Homework Statement A function f is defined on the whole of the xy-plane as follows: f(x,y) = 0 if x=0 f(x,y) = 0 if y = 0 f(x,y) = g(x,y)/(x^2 + y^2) otherwise a) g(x,y) = 5x^3sin(y) b) g(x,y) = 6x^3 + y^3 c) g(x,y) = 8xy For each of the following functions g determine if the...

Is there a profound relationship between Principle of Least Action and Continuity Equation? Can we derive one from another?

I'm struggling with the concept of uniform continuity. I understand the definition of uniform continuity and the difference between uniform and ordinary continuity, but sometimes I confuse the use of quantifiers for the two. The other problem that I have is that intuitively I don't...

Okay so the question is: Let $$f:R^2 \rightarrow R$$ by $$f(x) = \frac{x_1^2x_2}{x_1^4+x_2^2}$$ for $$x \not= 0$$ Prove that for each $$x \in R$$, $$f(tx)$$ is a continuous function of $$t \in R$$ ($$R$$ is the real numbers, I'm not sure how to get it to look right). I am letting $$t_0...

For every interval [ f(a)-e, (fa)+e ] there exists an interval [ f(a-d), f(a+d) ] such that [ f(a)-e, (fa)+e ] includes [ f(a-d), f(a+d) ] is this definition equivalent to the epsilon-delta definition?

Hi, Consider a vertical relatively long cylinder of constant radius open at both ends. We fill this cylinder with water and prevent water from falling down by a certain sheet as seen in the figure. Now suppose we remove the sheet suddenly. Let v1 be the speed of the upper surface of...

Hello. This is an improvement on a previous post, "Continuity of y^2". My original plan was to first prove that y and y^2 were continuous and then prove by induction that y^n was continuous; however, in the process of doing so I think I found a better way. This proof is for rudimentary practice...

I wrote up a proof for the continuity of y^2 for practice. Is this acceptable in the context of a Real Analysis I course? QED Thanks!

So right now I am reading on continuity in topology, which is stated as a function is continuous if the open subset of the image has an open subset in the inverse image... that is not the issue. I just read the pasting lemma which states: Let X = A\cupB, where A and B are closed in X. Let...

Homework Statement Prove ## f(x,y,z)=xyw## is continuos using the Lipschitz condition Homework Equations the Lipschitz condition states: ##|f(x,y,z)-f(x_0,y_0,z_0)| \leq C ||(x,y,z)-(x_0,y_0,z_0)||## with ##0 \leq C## The Attempt at a Solution...

Greetings, In Griffiths E&M, 3rd. Ed., on page 214, the following is part of the derivation of the continuity equation (the same derivation is shown on the Wikipedia article for the current density, under the continuity equation section: http://en.wikipedia.org/wiki/Current_density)...

Homework Statement . Prove that a metric space X is discrete if and only if every function from X to an arbitrary metric space is continuous. The attempt at a solution. I didn't have problems to prove the implication discrete metric implies continuity. Let f:(X,δ)→(Y,d) where (Y,d) is...

I don't really understand this question... I'm given a graph with x approaching and hitting 1, making y=2 (filled dot). Then there's a discontinuity jump at (1,3) which is the empty dot, then there's 2 other points on this small curve with empty dots at (3,4.5) and at (5,4), then another jump...

Homework Statement so a function was only continuous if and only if lim x ---> a = f(a) but I read some other books saying a functions continuity can only be judged based on their allowed domains. Does that mean the first definition fails if the domains of where the rule fails are...

Homework Statement f(x) = [1 - tan(x)]/[1 - √2 sin(x)] for x ≠ π/4 = k/2 for x = π/4 Find the value of k if the function is continuous at x = π/4 The Attempt at a Solution This means that lim x → π/4 f(x) = k/2 I put x = (π/4 + h) and then...

Homework Statement f(x) = sin ∏x/(x - 1) + a for x ≤ 1 f(x) = 2∏ for x = 1 f(x) = 1 + cos ∏x/∏(1 - x)2 for x>1 is continuous at x = 1. Find a and b Homework Equations For a lim x→0 sinx/x = 1. The Attempt at a Solution I tried...

How "continuity" of a map Τ:M→M, where M is a Minkowski space, can be defined? Obviously I cannot use the "metric" induced by the minkowskian product: x\cdoty = -x^{0}y^{0}+x^{i}y^{i} for the definition of coninuity; it is a misinformer about the proximity of points. Should I use the Euclidean...

I was thinking of a pathological function that, according to my intuitive ideas, would be discontinuous, but it actually satisfies a certain kind of continuity. First I claim that any element x∈[0,1) can be expressed in its decimal [or other base] expansion as x=0.d1d2d3... Where each di is an...

Homework Statement For which values of a E ℝ, is the function given by f(x) = pieceise function x^2+4x-4, x<a cos((x-a)/2) , x ≥ a. continuous at x=a Homework Equations I'm getting stuck on the algebra part to be honest. The Attempt at a Solution lim x→a f(x)= f(a) to be...

Homework Statement Find continuity of function f(x)= (x^2-1)/(x-1) at x = 1 Homework Equations limit f(x) as x-> = L The Attempt at a Solution i KNOW it can be easily solved by stating that at x = 1 function becomes infinity,so discontinous it is actually...But as we do in finding domain...

Does anyone happen to know who wrote down the first continuity equation and with regard to what? I know it shows up everywhere but was it originally a fluid dynamics equation? I've been trying to research this but I'm not coming up with much history on it. Thanks!

I've seen many definitions of continuous functions. They all describe x in a domain, but there's not really much explanation about the domain considerations beyond examples with "all the reals" and "an interval of the reals." I'm trying to figure out what continuity would mean on a missing...

What is the difference between Lipschitz continuous and uniformly continuous? I know there different definitions but what different properties of a function make them one or the other(or both). So Lipschitz continuity means the functions derivative(gradient) is bounded by some real number and...

Hello everybody! Given a topological space ##X## and two functions ##f,g:X\rightarrow \mathbb{R}##, it is rather easy to prove that ##x\rightarrow \max\{f(x),g(x)\}## is continuous. I wonder if this also holds for infinitely many functions. Of course, the maximum doesn't need to exist, so we...