Continuity Definition and 876 Threads

I've been thinking... Since derivatives can be written as: f'(c)= \lim_{x\rightarrow{c}}\frac{f(x)-f(c)}{x-c} and for the limit to exist, it's one sided limits must exist also right? So if the one sided limits exist, and thus the limit as x approaches c (therefore the derivative at c)...

Are projections always continuous? If they are, is there simple way to prove it? If P:V->V is a projection, I can see that P(V) is a subspace, and restriction of P to this subspace is the identity, and it seems intuitively clear that vectors outside this subspace are always mapped to shorter...

Homework Statement how do you show that sin(1/x) is continuous on (0,1)? (i know it's also continuous on (0, infinite)). Homework Equations The Attempt at a Solution |f(x)-f(xo)| = |sin(1/x)- sin(1/xo)|= |2sin((xo-x)\2)cos((xo+x)/2)| =< 2|sin((xo-x)/(2xox))|=<...

Hi guys. Final tomorrow and i had some last minute questions for proving/disproving a function is uniformly cont. Basically i want to know if the following proofs are acceptable Consider f(x)=1/x for x element (0, 2) = I Proof 1: f(x) does not converge uniformly on I. In order for...

In my analysis class we were posed the following question: Give an example of a uniformly continuous function f: (0,1) ---> R' such that f' exists on (0,1) and is unbounded. we came up with the example that f(x) = x*sin(1/x) if you interpret the question to mean f' is unbounded, not f...

Define h : \mathbb{R} \rightarrow \mathbb{R} h(x) = \begin{cases} 0 &\text{if\ }\ x \in \mathbb{Q}\\ x^3 + 3x^2 &\text{if\ }\ x \notin \mathbb{Q} \end{cases}. a.) Determine at what points h is continuous and discontinuous. Prove results. b.) Determine at what points h is...

So, I know the proof for a non-decreasing set using the continuity property, and I'm wondering if I have to use the intersection of all pairwise disjoint sets rather than the union, as seen in the non-decreasing proof. Any help would be greatly appreciated!

[SOLVED] Continuity on a piece-wise function Problem: Suppose: f(x)=\left\{\begin{array}{cc}x^2, & x\in\mathbb{Q} \\ -x^2, & x\in\mathbb{R}\setminus\mathbb{Q}\end{array}\right At what points is f continuous? Relevant Questions: This is in a classical analysis course, not a...

Homework Statement If lim x--> a of [f(x) + g(x)]=2 and lim x--> a of [f(x) - g(x)] = 1, then find lim x--> a f(x)g(x) Homework Equations Theorems of continuity The Attempt at a Solution Since I'm not quite sure if what I began with was right, it didn't yield me any type of a...

Homework Statement The inside diameters of the larger portions of the horizontal pipe as shown in the image (attached) are 2.50 cm. Water flows to the right at a rate of 1.80*10^4 m^3/s. What is the diameter of the constriction. Homework Equations Continuity equation Rate of Volume...

Homework Statement Suppose that the function f is continuous on [a,b] and X1 and X2 are in [a,b]. Let K1 and K2 be positive real numbers. Prove that there exist c between X1 and X2 for which f(c) = (K1f(X1) + K2f(X2))/(K1+k2) Homework Equations The Attempt at a Solution I...

Suppose that we will proof the continuity of the first maxwell equation: So we have div(\vec{E})=\frac{1}{\epsilon _0} \rho than \iiint \ div(\vec{E}) = \oint_v \vec{E} d\vec{s}=\iiint \frac{1}{\epsilon _0 } \rho than follewed E_{y1} l -E_{y2}l=Q Therefore E must continue is this a...

Homework Statement If X is bounded non empty subset in R (usual) and f:X->R is uniformly continuous function. Prove that f is bounded. Homework Equations The Attempt at a Solution Since X is bounded in R, it is a subset of cell. And all cells in R are compact.All bounded sub...

Homework Statement Find all values of the parameter a>0 such that the function f(x)=\left\{\begin{array}{cc}\frac{a^x+a^{-x}-2}{x^2},x>0\\3ln(a-x)-2,x\leq0\end{array}\right The Attempt at a Solution \lim_{x\rightarrow 0}\frac{a^x+a^{-x}-2}{x^2}=0 0=3ln(a-0)-2\rightarrow...

Homework Statement If an odd function g(x) is right-continuous at x = 0, show that it is continuous at x = 0 and that g(0) = 0. Hint: Prove first that \lim_{x \to 0^{-}} g(x) exists and equals to \lim_{x \to 0^{+}} g(-x) Homework Equations The Attempt at a Solution Suppose...

Homework Statement The question is to find 2 functions (f(x) and g(x) let's say) such that they're both NOT continuous at point a but at the same time, f(x)+g(x) and f(x)g(x) are continuous. Homework Equations The Attempt at a Solution I was thinking of letting f(x) = x +...

It began with my trying to prove that a uniformly continuous function on a bounded subset of the line is bounded. I took the hard route cause I couldn't figure out how to do this directly. I prove that if a real function is uniformly continuous on a bounded set E then there exists a continuous...

Homework Statement 1) If f is a continuous mapping from a matric space X to metric space Y. A E is a subset of X. The prove that f(closure(E)) subset of closure of f(E). 2) Give an example where f(closure (E)) is a proper subset of closure of f(E). Homework Equations The...

Homework Statement Where is the function f(x) continuous? f(x) = x, if x is rational 0, if x is irrational Homework Equations The Attempt at a Solution Is this correct?: I approach some c =/= 0, 1st through x's that are rational and prove there...

1) Prove that lim x_k exsts and find its value if {x_k} is defined by k->inf x_1 = 1 and x_(k+1) = (1/2) x_k + 1 / (sqrt k) [My attempt: Assume the limit exists and equal to L then L= (1/2) L + 0 => (1/2) L = 0 => L=0 Now I have to prove that the limit indeed exists, I want to use the...

Homework Statement Prove that if f(x) is Holder continuous, i.e, \sup_{a<x , y<b} \frac{\abs{f(x) - f(y)}}{\abs{x-y}^\alpha} = K^f_\alpha<\inf with \alpha > 1 , then f(x) is a constant function Homework Equations The Attempt at a Solution I've been staring at this for a...

I've heard some people say that the wave function and its first derivative must be continuous because the probability to find the particle in the neighborhood of a point must be well defined; other people say that it's because it's the only way for the wave function to be physically significant...

I was trying to solve the practice problems in my textbook, but I am highly frustrated. The terrible thing is that my textbook has a few to no examples at all, just a bunch of theorems and definitions, so I have no idea how to solve real problems...I am feeling desperate... Note: Let x E...

Does the notion of continuity exist in modern algebra? If so how do they arise?

I need a push with the following theorem, thanks in advance. Let X and Y be normed spaces, and A : X --> Y a linear operator. A is continuous iff A is bounded. So, let A be continuous. Then it is continuous at 0, and hence, for \epsilon = 1 there exists \delta > 0 such that for all x from...

My problem is this. Let f:\mathbb{R}^{2}\longrightarrow \mathbb{R}^{2} be a continuous function that satifies that \forall q\in\mathbb{Q}\times\mathbb{Q} we have f(q)=q. Proof that \forall x\in\mathbb{R}^{2} we have f(x)=x. I have worked out that because it is continuous, f satisfies that...

I encountered the following problem in the defination of 'continuity of a function'. We check the continuity of a function in its domain. Consider a function f defined by f(x)=(x^2-4)/(x-2). Its domain is R-{2}. i.e. the continuity of the function will be checked in R-{2}. The...

Hello Guys According to Classical electrostatics, when you apply a voltage across a capacitor, +Q and -Q charges are induced on a delta region at the interface of the dielectric and the metal electrode. The electric field inside the dielectric is finite and constant while the electric field in...

Homework Statement We have a worksheet with practice final questions and I'm really stuck on this one on continuity: Suppose h: (0,1) -> R has the property that for all x in (0,1), there exists a delta>0 such that for all y in (x, x+delta)\bigcap(0,1), h(x) <= h(y) a) prove that if h...

Hi, it's been awhile since I have studied Lebesgue measure so I'm trying to re-learn the material on my own. Most of my friends don't remember much as well so it's been a bit of a struggle trying to work on these problems on my own. Thank you for any kind of help! OMIT Question 1. If...

Homework Statement Show that \Sigma (from n=1 to infinity) of sin(nx)/n^2 is continuous on R Homework Equations The Attempt at a Solution No idea, any help would be greatly appreciated.

A rule of thumb seems to be that a quantum equation converts to its classical correspondent by replacing h (Planck's constant) in the former with zero for the latter. What do you think about the possibility for a continuum of intervening equations, wherein h eventually decreases to zero? Would...

Homework Statement Prove that the function \int^{1}_{x}\frac{sin t}{t}dt is uniformly continues in (0,1). Homework Equations The Attempt at a Solution First if all, I defined f(x) as sin(x)/x for x=/=0 and 0 for x=0. So f is continues in [0,1]. Now G(x) = \int^{x}_{1}f(t) dt...

Hi there just a general question: this involves continuity and differentiability suppose: f(x) = sin 1/x if x not equal to 0 f(0) = 0 PROVE F IS NOT DIFFERENTIABLE AT 0 i understand if it is not differentiable at 0 then it may not be continuous at 0. however is there...

Homework Statement 1) f is some function who has a bounded derivative in (a,b). In other words, there's some M>0 so that |f'(x)|<M for all x in (a,b). Prove that f is bounded in (a,b). 2) f has a bounded second derivative in (a,b), prove that f in uniformly continues in (a,b)...

Homework Statement Prove that if y>=x>=0: a) y^2 arctan y - x^2 arctan x >= (y^2 - x^2) arctan x b) \ | \ y^2 arctan y - x^2 arctan x \ | \ >= (y^2 - x^2) arctan x c) use (b) to prove that x^2 arctan(x) isn't UC in R. Homework Equations The Attempt at a Solution a) We...

I have this one as well, using the continuity equation to explain how a jet engine provides a foward thrust for an airplane. I have the equation but can some one explain this to me in laymen's terms. \frac{\partial\rho\left(\vec{r},t\right)}{\partial...

I have this question that I have stumped on for quite some time, discuss how the continuity equation and bernoulli's equation might relate to one another? Please post if you can explain this to me? Thx

Homework Statement True or false: 1)If f is bounded in R and is uniformly continues in every finate segment of R then it's uniformly continues for all R. 2)If f is continues and bounded in R then it's uniformly continues in R. Homework Equations The Attempt at a Solution 1) If...

I am trying to understand the concept of limit for a single variable case. I don't understand why while defining a limit a function may not be defined at the point? For a function to be continous, why should we care about the existence of limit when we can define the function at every point on...

Can anyone tell me whether sin|x| and cos|x| is differentiable at x=0 ? As far as i know, cos(x) and sin(x) is differentiable at all x. If i try to solve this, lim h->0 (f(x+h) - f(x))/h when x=0, and substitute cos|x| for f(x). lim h->0 (cos|h| - cos|0|)/(h) = 1, so cos|x| is...

Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...

Homework Statement I'm new here and I would like to ask a simple Q: what is the physical meaning of the continuity equation from (electrodynamic 1) I mean it's related to the electromagnatic problems Homework Equations The Attempt at a Solution I know the answer in my language...

Homework Statement For reference, this is chapter 11, problem 12 of Rudin's Principals of Mathematical Analysis. Suppose |f(x,y)| \leq 1 if 0 \leq x \leq 1, 0 \leq y \leq 1 ; for fixed x, f(x,y) is a continuous function of y; for fixed y, f(x,y) is a continuous function of x. Put g(x) =...

Homework Statement f(x) is a piecewise function defined as: |x-3| x>=1 \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} x<1 Discuss the continuity and differentiability of this funtion at x=1 and x=3 Homework Equations The Attempt at a Solution At x=3, this function is continuous...

Homework Statement A liquid with a specific gravity of 0.9 is stored in a pressurized, closed storage tank to a height of 7 m. The pressure in the tank above the liquid is 8700 Pa. What is the intial velocity of the fluid when a 5 cm valve is opened at a point 0.5 m from the bottom of the...

For non-fundamental functions obtained by a set of fundamental functions (either by multiplication, addition, division, compound or all together), and given those fundamental functions are all continuous on the desired intervals, will those non-fundamental functions also be continuous? I know...

Help! I am stuck on the following derivation: Use the conservation of mass to derive the corresponding continuity equation in cylindrical coordinates. Please take a look at my work in the following attachments. Thanks! =)

Theorem: Let f be continuous on [a,b]. The function g defined on [a,b] by http://tutorial.math.lamar.edu/AllBrowsers/2413/DefnofDefiniteIntegral_files/eq0051M.gif is continuous on [a,b], differentiable on (a,b), and has derivative g'(x)=f(x) for all x in (a,b) 1) Given that g is defined...

i have a continuous function f:R->R and we are given that lim f(x)=L as x approaches infinity and limf(x)=L as x approaches minus infinity, i need to prove that f gets a maximum or minimum in R. obviously i need to use weirstrauss theorem, but how to implement it in here. i mean by...