Continuity Definition and 876 Threads

Hi. In the book I'm reading it gives the function f(x) = 0, if x is irrational f(x) = 1/q, if x=p/q in lowest terms. It says this is continuous at all irrational x. This i can understand i think, because you can show that f(x) tends to zero, as x tends to a, for all a. For this you...

Homework Statement Suppose a function is continuous at a point, c. Does this mean there exists an interval around c which is also continuous? If so prove Homework Equations The Attempt at a Solution

Homework Statement Question Details: The question reads: Show that the equation: dA/A + dv/v + dρ/ρ = 0 applies to a one-dimensional steady flow. (Here 'one dimensional' means that both the density ρ and seed v = - v . n (vectors) are constant across any cross-sectional area A...

there is one problem. the problem is related with contuinity of afunction and i tried like as shown below.so if anyone who is intersted to help me i like .. the problem is prove that if f(a+b)=f(a)f(b) for all a and b ,then f is cntiniuous at every real number.here there is given information...

Hi, This may sound lame but I am not able to get the definition of uniform continuous functions past my head. by definition: A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta...

I'm going through a topology book (Introduction to Topology by Bert Mendelson.) In one of the first chapters the author defines continuity in an epsilon-delta manner (not limit definition.) Here is the definition: I'm confused because, if I understand correctly, we can set both \epsilon and...

Homework Statement If the continuous function f(x) has a derivative f'(x) at each point x in the neighborhood of x=\xi, and if f'(x) approaches a limit L as x \rightarrow \xi, then show f'(\xi) exists and is equal to L.Homework Equations The Attempt at a Solution Since the derivative exists...

Homework Statement Discuss the continuity of the function f defined for all x belongs to [0,1] by f(x)=x if x is rational and f(x)=x^2 is x is irrational. Homework Equations The Attempt at a Solution I have no idea how to begin this question...some help would be great thanks!

Suppose f:A-->R is monotone (ACR: reals) and suppose the range of f is an interval, show f is continuous on A. By drawing a picture, I can see the conclusion. Since f is monotone, the only type of discontinuity it may have is a jump discontinuity. But since the range of f is an interval...

Homework Statement A water line with an internal radius of 6.1*10^-3 m is connected to a shower head that has 24 holes. The speed of the water in the line is 1.2 m/s. (b) At what speed does the water leave one of the holes (effective radius = 4.6*10^-4 m) in the head...

Homework Statement f:Rn->Rn is continuous and satisfies |f(x)-f(y)|>=k|x-y| for all x, y in Rn and some k>0. Show that F has a continuous inverse. Homework Equations The Attempt at a Solution It is easy to show that f is injective, but I've no idea how to prove the surjectivity. I...

Hi there! :) I'm trying to understand a theorem, but it's full with analysis (or something) terms unfamiliar to me. Is there an intuitive interpretation for the sentence: 'An operator being limited is equivalent to continuity in the topolgy of the norm'? Also, how can I partially...

When doing some self-study in probability, I have seen a number of authors state, without proof or justification, that the inverse of a matrix is continuous. For instance, a passage in a popular econometrics text (White (2001)) reads: "The matrix inverse function is continuous at every point...

Homework Statement let f(x)= (x^2)/(1+x) for all x in [ifinity, 0) proof that f(x) is uniformly continuous. can anyone help me with this problem Homework Equations using the definition of a uniform continuous function The Attempt at a Solution i did long division to simplify the...

[b]1. Show that the cross product is a continuous function [b]3. I have tried to apply the definition of continuity: find a delta such that |x-y|< delta implies |f(x)-f(y)|< epsilon but I'm having trouble finding a delta that would take me to the conclusion.

[b]1. Show that the cross product is a continuous function. The Attempt at a Solution I have tried to apply the definition of continuity: find a delta such that |x-y|< delta implies |f(x)-f(y)|< epsilon but I'm having trouble making sense of what |x-y| is. As I see it, x is a pairs of...

Imagine a jet of fluid (perhaps air) impinging on a flat plate. It could be said that the jet has a slightly higher mean velocity in the direction normal to the flat surface (we'll arbitrarily call this X). From a classical thermodynamic point of view it could be said that the gas has a higher...

Hi all! I´m having some trouble finding a delta for f(x)=(x-2)² using the epsilon-delta definition for fixed epsilon and x_0. Here´s what I come up with: |f(x)-f(x_0)|<\epsilon...

From my textbook, this is the proof given for a theorem stating that any function continuous in a closed interval is automatically uniformly continuous in that interval. Proof: "If f were not uniformly continuous in [a, b] there would exist a fixed \epsilon > 0 and points x, z in [a, b]...

Homework Statement Show that f(x)=\frac{1}{x^{2}} is uniformly continuous on the set [1,\infty) but not on the set (0,1]. Homework Equations The Attempt at a Solution I've been working at this for at least 2 hours now, possibly 3, and I can't say I really have much of any idea...

Homework Statement Suppose f: [0,1] -> [0,1] is such that f attains each of its values exactly twice Show that f cannot be continuousThe Attempt at a Solution I assumed that f is continuous and tried to break it up into cases and show that there must be a value that is obtained 3 times. since...

Ok, let's say I had 3x^{2}-2x+1 I know we have lx-2l<\delta Also l(x-2)(3x+4)l<\epsilon My problem with these types of questions is dealing with the l3x+4l. I just don't really know what to do.

Homework Statement suppose f and g are uniformly continuous functions on X and f and g are bounded on X, show f*g is uniformly continuous. The Attempt at a Solution I know that if they are not bounded then they may not be uniformly continuous. ie x^2 and also if only one is bounded...

Homework Statement if f and g are 2 uniformly continuous functions on X --> R show that f+g is uniformly continuous on X The Attempt at a Solution I tried showing that f+g is Lipschitz because all Lipschitz functions are uniformly continuous. So i end up with d(x_1,x_2) <...

Homework Statement If f and g are uniformly continuous on X, give an example showing f*g may not be uniformly continuous. The Attempt at a Solution i think if the functions are unbounded the product will not uniformly continuous. Is there a specific example of this function..?

Homework Statement f(x)=e^x g(x)=lnx h(x)=x^x=e^xlnx If f and g are continuous prove h(x) is continious for x>0 Homework Equations The Attempt at a Solution Ok this confuses me, because I would think that it wouldn't be too bad too do if h(x)=f(g(x)). Maybe the book had a typo?

Homework Statement Let p and q be a polynomial and x0 be a zero of q of multiplicity m. Prove that p/q can be assigned a value at x0 such that the function thus defined will be continuous there iff x0 is a zero of p of multiplicity greater than or equal to m. Homework Equations The...

Homework Statement 1. show there is some point x in the interval [0,pi/2] so that x = cos(x)^2 2. let f:R-> be continuous at c and suppose f(c) =1. show that there is some a > 0 such that f(x) > 1/2 whenever |x-c| < a Homework Equations intermediate value theorem. maximum minum...

Homework Statement f(x) = x^2 - 4x + a g(x) = \lim_{n\rightarrow\infty} \frac {2|x-b|^n + 1}{|x-b|^n + 1} let h(x) = f(x)g(x) Find the sum of a+b that makes h(x) continuous for all x. Homework Equations Power Series? Derivation to test continuity? The Attempt at a Solution...

Homework Statement Let f: [0,1] -> R (R-real numbers) be a continuous non constant function such that f(0)=f(1)=0. Let g_n be the function: x-> f(x^n) for each x in [0,1]. I'm trying to show that g_n converges pointwise to the zero function but NOT uniformly to the zero function...

Homework Statement Given that f(x + y) = f(x) + f(y), prove that (a) if this function is continuous at some point p, then it is continuous everywhere (b) this function is linear if f(1) is continuous. Homework Equations definition of continuity The Attempt at a Solution (a) I...

Homework Statement Suppose f is continuous on [0,2]and thatn f(0) = f(2). Prove that there exists x,y in [0,2] such that |y-x| = 1 and f(x) = f(y) Homework Equations The Attempt at a Solution I got the following 1 line proof. Suppose g(x) = f(x + 2) - f(x) on I = [0,2]...

Homework Statement Given the Hamiltonian H=\vec{\alpha} \cdot \vec{p} c + mc^2 = -i \hbar c \vec{\alpha} \cdot \nabla + mc^2 in which \vec{\alpha} is a constant vector. Derive from the Schrödinger equation and the continuity equation what the current is belonging to the density \rho...

Homework Statement Show that if a function f:(0,1) --> lR is uniformly continuous, f is bounded. Homework Equations - The Attempt at a Solution Really don´t know. I started thinking about Weierstrass Thereom but I am not sure that it´s ok. Now I think that may be is something...

Homework Statement y= y= {1+3ax+2x^2} if x is < or = 1 {mx+a} if x>0 what values for m and a make x continuous and differentiable at 1? Homework Equations n/a The Attempt at a Solution i solved for when x=1. i got 3+3a. this is also the right hand...

X, Y metric spaces. f:X-->Y and X is compact. How do I prove that f is continuous if and only if G(f)={(x,f(x)):x in X} C X x Y is compact. I think for the forward direction, since f is continuous and X is compact, then f(X) is compact. Hence, G(f)=X x f(X) is compact as a cross product of...

Let I = [0,1] be the closed unit interval. Suppose f is a continuous mapping from I to I. Prove that for one x an element of I, f(x) = x. Proof: Since [0,1] is compact and f is continuous, f is uniformly continuous. This is where I'm stuck. I'm wondering if I can use the fact that since...

Homework Statement Let V and V' be real normed vector spaces and let f be a linear transformation from V to V'. Prove that f is continuous if V is finite dimensional. The attempt at a solution Let v_1, v_2, \ldots, v_n be a basis for V, let e > 0 and let v in V. I must find a d such that...

Homework Statement Let (E, m) and (E', m') be metric spaces, let A and B be closed subsets of E such that their union equals E, and let f be a function from E into E'. Prove that if f is continuous on A and on B, then f is continuous on E. The attempt at a solution I have approached this...

Homework Statement Prove that if f(x) satisfies the functional equation f(x+y) = f(x) + f(y) and if f is continuous then f(x) = cx for some constant c. Homework Equations N/A The Attempt at a Solution Assume |f(a)| > |ca| for some a in the domain of f. Since f is continuous at...

(Problem 62 from practice GRE math subject exam:) Let K be a nonempty subset of \mathbb{R}^n, n>1. Which of the following must be true? I. If K is compact, then every continuous real-valued function defined on K is bounded. II. If every continuous real-valued function defined on K is...

Dear all, I would appreciate if you could help me with the following problem: A person is standing still on a 2D environment and let's assume that its initial position Xo is given. The person is moving by applying a force function over time say f(t). As a result, using numerical integration we...

The question seemed simple enough, but something feels funny about my proof. I would appreciate if someone could please check it. Question: Prove that if f(x) is monotonic on [a,b] and satisfies the intermediate value property, then f(x) is continuous. Proof: Let e denote epsilon and d denote...

A function f:D\rightarrowR is called a Lipschitz function if there is some nonnegative number C such that absolute value(f(u)-f(v)) is less than or equal to C*absolute value(u-v) for all points u and v in D. Prove that if f:D\rightarrowR is a Lipschitz function, then it is uniformly...

Please consider a U-tube filled with an incompressible fluid as in the attached figure. Piston P divides the fluid in two segments. When P moves, the fluid particles on immediate vicinity of either face (points marked 1 and 2) will have same velocity. Does this mean, they may considered to be...

The question asks to find a value for a and b that makes f continuous everywhere. f(x)= \frac{x - 4}{x-2} , where x<2 ax2 - bx + 3 , where 2<x<3 2x - a +b , where x > or = 3 I know that in order for a function to be continuous the limit as x approaches 2 must be equal from...

does eqn of continuity apply to only incompressible fluids?is there an eqn for compressible fluids?

Homework Statement Let a function f : R => R be convex. Show that f is necessarily continuous. Hence, there can be no convex functions that are not also continuous. Homework Equations The Attempt at a Solution F is continuos if there exist \epsilon >0 and \delta>0 such that |x-y|<...

It is true that \frac{\partial}{\partial x^\beta} T^{0 \beta} = \gamma^2 c \left( \frac{\partial \rho}{\partial t} + \vec{\nabla} \bullet \left[ \rho \vec{v} \right] \right) = 0 but, how do we arrive at this point? What is in T^{ \alpha \beta} and how do we compute it for any...

Homework Statement We have a piecewise continuous function and T-periodic function f and we have that: F(a) = \int_a^{a + T} {f(x)dx} I have to show that F is diferentiable at a if f is continuous at a. My attempt so far: I have showed that F is continuous for all a. If we look at one...