Continuous Definition and 1000 Threads

Let X be a compact space, (Y,p) a compact metric space, let F be a closed subset of C(X,Y) (the continuous functions space) (i guess it obviously means in the open-compact topology, although it's not mentioned there) which satisifes: for every e>0 and every x in X there exists a neighbourhood U...

Homework Statement How is the theorem "Continuous functions have closed graphs" proven in the setting of a general topological space? (assuming the theorem is still valid?)

Is there a continuous function f(x) defined on (-\infty,+\infty) such that f(f(x))=e^{-x}? My opinion is "no", and here is how i think: first of all if such a function exists, it should be a "one-to-one" function, that is for every y>0, there should be exactly one x such that f(x)=y. Thus by...

Dear friends, I just joined the forums, and I'm looking forward to being a part of this online community. This semester, I signed up for Analysis II. I'm a math major, so I should be able to understand pretty much everything you say (hopefully). However, I'd really appreciate it if you try...

Dear friends, I just joined the forums, and I'm looking forward to being a part of this online community. This semester, I signed up for Analysis II. I'm a math major, so I should be able to understand pretty much everything you say (hopefully). However, I'd really appreciate it if you try...

[SOLVED] The continuous dual is Banach Homework Statement I'm trying to show that the continuous dual X' of a normed space X over K = R or C is complete. The Attempt at a Solution I have shown that if f_n is cauchy in X', then there is a functional f towards which f_n converge pointwise...

I'm having trouble with the third part of a three part problem (part of the problem is that I don't even see how what I'm trying to prove can be true). The problem is: Let X and Y be topological spaces with X=E u F. We have two functions: f: from E to Y, and g: from F to Y, with f=g on the...

Homework Statement This is not a homework question. I am solving this from the lecture notes that one of my friend's has got from last year. If C(X) denotes a set of continuous bounded functions with domain X, then if X= [0,1] and fn(x) = x^n. Does the sequence of functions {fn} closed ...

Homework Statement Suppose that fk : X to Y are continuous and converge to f uniformly on every compact subset of the metric space X. Show that f is continuous. (fk is f sub k) Homework Equations Theorem from p. 150 of Rudin, 3rd ed: If {fn} is a sequence of continuous functions on E...

Let f(x) = \frac{x}{x-1}. Prove f(x) is uniformly cont. on the interval [1.5,\infty)

Now that I have finished with the 7 chapters of rudin i plan to study Part 1 of royden's real analysis. Here is a problem regarding borel sets Let f be a real valued function defined for all reals. Prove that the set of points at which f is continuous can be written as a countable...

Homework Statement Okay, so if f and g are continuous functions at a, then prove that f/g is continuous at a if and only if g(a) # 0 Homework Equations Assuming to start off the g(a)#0, by the delta-epsilon definition of continuity, basically, We know that |f(x)| and |g(x)| are bounded...

Does the ring of continuous functions over the real numbers have no zero divisors? If no 0 divisor, how can I prove it? Else, what is a counter example?

Experiment with a family of paths with common endpoints, say z(t) = t + \iota a sin(t) 0 \leq \ t \ \leq \pi, with real parameter a. Integrate non-analytic functions (Re(z), Re(z^2), etc.) and explore how the result depends on a. Take analytic functions of your choice compare and comment...

Homework Statement A random variable has distribution function F(z) = P(y<= z) given by (this is a piecewise function) f(z) = 0 if z < -1 1/2 if -1 <= z < 1 1/2 + 1/4(z-1 if 1 <= z < 2 1 if 2 <= z What is P(Y = 2)? Find all the numbers t with the property that both P(Y <= t) >=...

Homework Statement suppose f is a function with the property f(x+y)=f(x)+f(y) for x,y in th reals. suppose f is continuous at 0. show f is continuous everywhere. Homework Equations The Attempt at a Solution I do not understand how to show that f is continuous everywhere.

Problem statement Given f:[1,2]->R defined by f(x) = x^2. Show that this function is continuous Problem Solution (my version at least) 1- It is known sequence {a[n]b[n]} converges to {ab} 2- Definition of continuous: if every sequence {cn} in f we have f(cn) -> f(c) 3- Given our domain...

say function f is continuous on (-\infty,\infty). show that f can be written as f = g + h, where g is an even function and h is an odd function. help pleaseee!

I'm confused about a point that my book on real analysis is making about the difference between the definition of a function being continuos at at point and the definition of a function having a limit L at a point. My rough understanding of the matter is that in order for a function to be...

Homework Statement In topology, a f: X -> Y is continuous when U is open in Y implies that f^{-1}(U) is open in X Doesn't that mean that a continuous function must be surjective i.e. it must span all of Y since every point y in Y is in an open set and that open set must have a pre-image...

Homework Statement A kind person is helping me to self-study some aspects of topology, continuity, etc. He posed the following exercise for me, which I can't do, but he doesn't have time to write up the full solution. Ex: Show that a mapping f between Hausdorff spaces is continuous if...

OKKK, f(x) = 2x + 1, for x =< (greater than or equal to) 2 .5x^2 + k for x > 2 --- FOR what value of k will f be continuous ? MOST IMPORTANTLY, if k=4, is f differentiable at 2?

I Apologize in advance for the amount of questions i plan on asking you guys this year. Real Analysis is my first upper division math class and i have not trained my mind to think abstractly enough yet. Homework Statement i) Show that f(x) = x^3 is continuous on R by using...

T is a linear transformation from R^m->R^n, prove that T is continuous. I have proved that there's always a positive real number C that |T(x)|<=C|x|. How shall I proceed then? Thanks~

Is it true that f is a continuous function from A \times B to C (A, B, C are topological spaces) if and only if f_{a}: \{a\}\times B \longrightarrow C and f_{b}: A\times \{b\} \longrightarrow C are continuous for all a\in A, b\in B ? f_a(b) = f_b(a) = f(a,b)

Homework Statement We recently proved that if a function, f, is continuous, it's absolute value |f| is also continuous. I know, intuitively, that the reverse is not true, but I'm unable to come up with an example showing that, |f| is continuous, b f is not. Any examples or suggestions would...

This is not a homework but it is a question in my mind.please guide me. Let X and Y be topological spaces,let f : X -----> Y is a function. when the following statements are equivalent?: 1) f is continuous 2) f(A') is subset of f(A)' ,for every A subset of X. Symbols: A' i.e...

Homework Statement 380g of a liquid at 12'C in a copper calorimeter weighing 90g is heating at a rate of 20 watt for exactly 3 minutes to produce a temperature of 17'C. If the specific heat capacity of copper is 40Jkg-1K-1, the thermal capacity of the heater is negligible, and there is a...

I am a student in a DMS program. Our instructor poised the following question (worth extra credit!) if we can answer it and back it up. OPERATING FREQUENCY IN PULSED WAVE TRANSDUCERS IS DETERMINED BY: A. FREQUENCY OF THE VOLTAGE B. PULSE REPITITION FREQUENCY C. THE MEDIUM ONLY D. THE THICKNESS...

Hello need help with this one. f:[0,1] --> [0,1] f( .x1 x2 x3 x4 x5 ...) = .x1 x3 x5 x7 ( decimal expansion) prove that f is nowhere diffrentiable but continuous. i tried by just picking a point a in [0,1] and the basic definiton of differentiability about that point...doesnt seem...

Hi all, There are some dificult problems with discrete argument n that will be very easy if I can change it to continuous argument x. But I do not know what is the condition for that. For example: to calculate the sum of a1+a2 +a3+...an. when n goes to infinity, can I make it as S=integral...

If white light were emitted from Earth's surface and observed by someone in space, would its spectrum be continuous? Explain.

A rod of length 80 cm has a uniform linear charge density of 5 mC/m. Determine the Electric Field at a point P located at a perpendicular distance 57 cm along a line of symmetry of the rod i don't know what i wrong..but here is what i am doing linecharge(change in lenght)= change in Q...

Homework Statement 4.8 Show the following continuous theorem for sequences: if a_n \rightarrow L and f is a real valued function continuous at L, then bn = f(a_n) \rightarrow f(L). Homework Equations No real relevant equations here. Just good old proof I'm thinking. The Attempt at a Solution...

How do I show that f(x)=sum (from n=0 to infinity) cos(nx)e^-nx is a continuous function? x is from (0, infinity) So, I need to show that the series converges uniformly. I'm trying to say that |cos nx e^-nx| <= |e^-nx| and use Weierstrass comparison, but I can't find a function M_n to use for...

Hi all, why a wave function has to be a continuous function?

[Solved]Continuous random variable={-b-(b^2-4ac)^.5}/(2a)=x? Homework Statement A factory is supplied with grain at the beginning of ea week.The weekly demand,X thousand tonnes for grain from this factory is a continuous random variable having the probability density function given by...

Can you guys help me prove: Given a continuous and differentiable function (or surface) f: R^2 -> R, such that f(x,y) = z ... contour lines can always be drawn... the function is NOT bijective. I've been thinking of choosing any arbitrary point and showing that the curves that intersect to...

Hi, just a home work question I am having problems with. it has to be solved graphically. (1) Give an example of a continuous function f : [a,b] -> R that has no local maximum or local minimum at an endpoint

Rough idea behind integration is to sum lot's of small numbers (close to zero). Some problems lead to situations where you have to multiply lot's of numbers close to one. Is there any general theory of such products? Important results or tools?

I have an equation; f(x) = \sum_{i=1}^{x-1} s^i Where s is a constant. Is it possible to transform f(x) into continuous functions ? If so, how ?

Homework Statement Prove that if f is continuous at a, then so is |f| Homework Equations The Attempt at a Solution I know lim f = L x->a Not sure really where to go from here.

I'm a little nervous about a test I have on Thursday and I was wondering if this is adequate for a proof of the following equation. Homework Statement Show that f(x) is continuous at a=4 Homework Equations f(x) = x^2 + \sqrt{7-x} The Attempt at a Solution For f(x) to be continuous at a = 4...

Homework Statement f: [0,+\infty) \to \mathbb{R}: y \mapsto \int_0^{+\infty} y \arctan x \exp(-xy)\,dx. Show that this function is continuous in y if y \neq 0 and discontinuous if y = 0 Homework Equations The Attempt at a Solution I just can't get started, any hint?

i need to find the cardinality of set of continuous functions f:R->R. well i know that this cardinality is samaller or equal than 2^c, where c is the continuum cardinal. but to show that it's bigger or equals i find a bit nontrivial. i mean if R^R is the set of all functions f:R->R, i need to...

Homework Statement I've to show that if f:R->R is continuous in x' then f is limited in a suitable environment of x'. 2. The attempt at a solution My lecturer said we should use the following inequality |f(a)|=<|f(a)-f(y)|+|f(y)| But how should I go on, I know I have to show...

Give an example of a set X with two topologies T and S such that the identity function from (X,T) to (X,S) is continuous but not homeomorphic. I always struggle with these because I get overwhelmed by the generality that it has. Any ideas would be very much appreciated.

After reading http://en.wikipedia.org/wiki/Weierstrass_function it occurred to me that I could do the same thing to an integral: \int \sum_{i=0}^\infty \frac{sin(\frac{x}{3^i})}{2^i} dx = \sum_{i=0}^\infty \int \frac{sin(\frac{x}{3^i})}{2^i} dx = \sum_{i=0}^\infty...

Hi All! Preliminaries: Let H denote the Hilbert-space, and let A be a densely defined closed operator on it, with domain $D(A) \subset H$. On D(A) one defines a finer topology than that of H such way that f_n->f in the topology on D(A) iff both f_n->f and Af_n->Af in the H-topology. Let...

[FONT="Georgia"]Homework Statement Discuss the motion of a continuous string when the initial conditions are q'(x,0) = 0 and q(x,0) = Asin(3πx/L). Resolve the solution into normal modes. Show that if the string is driven at an arbitrary point, none of the normal modes with nodes at the driving...