Continuous Definition and 1000 Threads

Homework Statement Why is it that continuous functions do not necessarily preserve cauchy sequences. Homework Equations Epsilon delta definition of continuity Sequential Characterisation of continuity The Attempt at a Solution I can't see why the proof that uniformly continuous...

Homework Statement I'm given the pdf and asked to find F(y)/ cdf. I've calculated it many times, but I'm not getting the right numbers. the pdf is f(y)= .5, ....-2≤y≤0 .75-.25y,...1≤y≤3 0,...elsewhere so that means f(y)= 0,...y< -2 0.5, ...-2≤y≤0...

As in my previous thread we had: "Let f a function which satisfies $$|f(x)|\leq|x| \forall{x\in{\mathbb{R}}}$$ Proof that is continuous at 0. We concluded that since f(0)=0 then we found a delta equal epsilon so $$|f(x)|≤|x|<ϵ$$. But now I have: $$\textrm{g continuous at 0 and...

I don't see the subtle differences between continuous and uniformally continuous functions. What can continuous functinons do that unifiormally continuous functions can't or vice versa?

Homework Statement Homework Equations The Attempt at a Solution I understand that all i need to do is plug these two points into the formula and subtract to get the correct area, but i am not provided a mean or variance as i normally am, so I'm at a loss.

Homework Statement Suppose that f : ℝ→ℝ is continuous on ℝ and that lim f =0 as x→ -∞ and lim f =0 as x→∞. Prove that f is bounded on ℝ and attains either a maximum or minimum on ℝ. Give an example to show both a maximum and a minimum need not be attained. The Attempt at a Solution...

Homework Statement Let f be continuous on the interval [0,1] to ℝ and such that f(0) = f(1). Prove that there exists a point c in [0,1/2] such that f(c) = f(c+1/2). Conclude there are, at any time, antipodal points on the Earth's equator that have the same temperature. Homework Equations...

Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$ I know this has to do with the...

Let X be some topological space. Let A be a subspace of X. I am thinking about the following: If f is a cts function from X to X, and g a cts function from X to A, when is the piece-wise function h(x) = f(x) if x is not in A, g(x) if x is in A continuous? My intuition tells me they must agree...

Homework Statement Let g: ℝ→ℝ satisfy the relation g (x+y) = g(x)g(y) for all x, y in ℝ. if g is continuous at x =0 then g is continuous at every point of ℝ. Homework Equations The Attempt at a Solution Let W be an ε-neighborhood of g(0). Since g is continuous at 0, there is...

When f maps E into a metric space Y: (E is subset of metric space X) Is it eqivalent to say that f is a continuous mapping and that for a subset E of X, to say that for every p element of E, f is continuous at p.? thank you

Cardinality of the Preimage f^{-1}(y) of f:X-->Y continuous? Hi, All: Let X,Y be topological spaces and f:X-->Y non-constant continuous function. I'm curious as to whether it is possible for the fiber {f^{-1}(y)} of some y in Y to be uncountable, given that the fiber is discrete (this...

Homework Statement Define f:[-1,∞]→ℝ as follows: f(0) = 1/2 and f(x) =[(1 + x)^(1/2) - 1]/x , if x ≠ 0 Show that f is continuous at 0. Homework Equations Definition. f is continuous at xo if xoan element of domain and lf(x) - f(xo)l < ε whenever lx - xol < δ The Attempt at...

Can I use the definition of continuity of function from Baby Rudin, setting X as empty set? Rudin does not specify X is a non-empty set but he supposes p is in X. Anyway if I use it for empty set X, then is a function with a domain E which is a subset of X continuous at p? One more...

I'm trying to show that continuous f : [a, b] -> R implies f uniformly continuous. f continuous if for all e > 0, x in [a, b], there exists d > 0 such that for all y in [a, b], ¦x - y¦ < d implies ¦f(x) - f(y)¦ < e. f uniformly continuous if for all e > 0, there exists d > 0 such that for...

Homework Statement Let f_{n}(x)=\frac{-x^2+2x-2x/n+n-1+2/n-1/n^2}{(n ln(n))^2} Prove f(x) = \sum^{\infty}_{n=1} f_{n}(x) is well defined and continuous on the interval [0,1]. Homework Equations In a complete normed space, if \sum x_{k}converges absolutely, then it converges.The Attempt at...

or is space itseld quantized meaning an object moving from a to b would in some way look like A \/\/\/\/\0/\/\/\/ B or is it smooth and continuous like A--------0------B ?

A function defined on ℝ is continuous at x if given ε, there is a δ such that |f(x)-f(y)|<ε whenever |x-y|<δ. Does this imply that f(x+δ)-f(x)=ε? The definition only deals with open intervals so i am not sure about this. If this is not true could someone please show me a counter example for it...

Homework Statement assume a function F(x)=(a|x|^(a-1))*(sin(1/x))-((|x|^a)/(x^2))*(cos(1/x)) for x not equal to 0 F(x)=0 for x equal to 0 for what values of a that this function is continuous on R(real number) Homework Equations the F(x) is the differentiation of |x|^a sin(1/x)...

function f:R->R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。then if f is continuous then f1 and f2 may be chosen continuous, and if f is differentiable then f1 and f2 can be chosen differentiable i am quiet confusing this statement , if f1 is continuous f2 is not how their...

Homework Statement function f:R->R can be written as a sum f=f1+f2 where f1 is even and f2 is odd。show that if f is continuous then f1 and f2 may be chosen continuous, and if f is differentiable then f1 and f2 can be chosen differentiable 2. The attempt at a solution i have try some...

Homework Statement Let f : R -> R be a continuous function such that f(0) = 0. If S := {f(x) | x in R} is not bounded above, prove that [0, infinity) ⊆ S (that is, S contains all non-negative real numbers). Then find an appropriate value for a in the Intermediate Value theorem...

I need help on alternative motor that I can use for vacuum that can run continuously for 20 hours a day and it should be 12VDC powered. Basically I am doing a heat exhaust system for a data center. I want to use small pipes so it won't be bulky about 1.5" and for that to work I think strong...

[topology] "The metric topology is the coarsest that makes the metric continuous" Homework Statement Let (X,d) be a metric space. Show that the topology on X induced by the metric d is the coarsest topology on X such that d: X \times X \to \mathbb R is continuous (for the product topology on X...

Homework Statement a) Let {s_{n}} and {t_{n}} be two sequences converging to s and t. Suppose that s_{n} < (1+\frac{1}{n})t_{n} Show that s \leqt. b) Let f, g be continuous functions in the interval [a, b]. If f(x)>g(x) for all x\in[a, b], then show that there exists a positive real z>1 such...

Hi All, just a question regarding continuous functions. From what I understand if x > 2, then any value of 'a' should make this function continuous? Any clarification would be very helpful! Thanks in advance!

On the average, which provides more information: yes/no answers or answers neither yes nor no? Just asking.

Homework Statement Let f and g be two continuous functions defined on R. I'm looking to prove the fact that if they agree on Q, then f and g are identical. Homework Equations The Attempt at a Solution I'm not really sure where to start with this. Can someone point me in the right...

1. Homework Statement Find the total charge on a circular disc of radius ρ = a if the charge density is given by ρs = ρs0 (e^−ρ) sin2 φ C/m2 where ρs0 is a constant. Are the two limits of integration from 0 -> a for ρ and 0->2∏ for φ? In the example given in the notes, ρ varies, instead of...

Homework Statement Find the total charge on a circular disc of radius ρ = a if the charge density is given by ρs = ρs0 (e^−ρ) sin2 φ C/m2 where ρs0 is a constant. Are the two limits of integration from 0 -> a for ρ and 0->2∏ for φ? In the example given in the notes, ρ varies, instead of...

Homework Statement I am having trouble understanding how \textit{Δ}\vec{E}\textit{ = k}_{e}\frac{Δq}{{r}^{2}} (where ΔE is the electric field of the small piece of charge Δq) turns into \vec{E}\textit{ = k}_{e}\sum_{i}\frac{{Δq}_{i}}{{{r}_{i}}^{2}} then into \vec{E}\textit{ =...

Homework Statement Homework Equations The Attempt at a Solution I can show the second one, i.e. 1/3 [sqrt(y) +1] and need help in showing the first one. Can anyone guide me?

Homework Statement f\rightarrowℝ, f(x,y)=\frac{x^{2}y}{x^{6}+y^{2}} where (x,y)\neq(0,0) and f(0,0)=0. Is the function continuous at (0,0)? The Attempt at a Solution I tried to find the limit at (0,0) so I put y=x into the function f and got the limit 0 when x\rightarrow0. Tthen I...

Estimate the decay heat rate in a 3000 MWth reactor in which 3.2% mU-enriched U02 assemblies are being fed into the core. The burned-up fuel stays in the core for 3 years before being replaced. Consider two cases: 1. The core is replaced in two batches every 18 months. 2. The fuel...

Hello everyone, This is probably a really newbie question and I apologise for it. So a continuous function is one that is differentiable with respect to its input parameters. What happens when the input parameters can only take discrete values? So, I guess the function can, of course, not...

Homework Statement f(x) = x^2 - c^2 \mbox{ if } x < 5 f(x) = cx+11 \mbox{ if } x \geq 5 Find the two values of c for which the function would be continuous. Homework Equations The Attempt at a Solution I set these two equations equal to each other, plug in the value 5...

A line of positive charge is formed into a semicircle of radius R = 57.8 cm, as shown in the figure below. (The figure is a semicircle above the x-axis with angle θ measured from positive y axis centred at the origin) The charge per unit length along the semicircle is described by the...

Hello, below I have the problem and solution typed in Latex. For the first part, I just want someone to verify if I am correct. For the second part, I need guidance in the right direction

Hello, I want to test gearboxes. How do you recommend I test the continuous torque of a gearbox and also the peak torque of the gearbox? I want to test to see the maximum torque the gearbox can handle. How do you suggest I go about doing so? What type of equipments or measuring devices...

This is not a homework problem, but it may as well be, so I thought I'd post it here. Homework Statement The function f:[0,1] \to \mathbb{R} given by f(x)=\sqrt{x} is continuous on a compact domain, so it is uniformly continuous. Prove that f is uniformly continuous directly (with a...

I'm trying to prove that if a function is continuous on [a,b] and smooth on (a,b) then there's a point x in (a,b) where f'(x) exists. The definition of smoothness is: f is smooth at x iff \lim_{h \rightarrow 0} \frac{ f(x+h) + f(x-h) - 2f(x) }{ h } = 0 . I'm starting with the simpler case...

Homework Statement I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces: 1) span{(1,0,i,1),(0,1,1,-i)}, 2) All constant functions in V over the interval [a,b] Homework Equations I know that for a subspace W of an...

Homework Statement Homework Equations I'm looking over in old midterm to prepare for a final and can't figure out what the correct answer is. No answers were ever given. I'm not cheating on anything, would just like to know what the correct answer is and why :) The Attempt at a...

Hi all, I was having some troubles with a practise question and thought I'd ask here. Given an r.v. X has a pdf of f(x) = k(1-x2), where -1<x<1, I found k to be 3/4. And I found the c.d.f F(x) = 3/4 * (x - x3/3 + 2/3) Now I have to find a value a such that P(-a <= X <= a) = 0.95. I thought...

Homework Statement Let f be a function such that: \left |f(u) - f(v) \right | \leq \left | u - v\right | for all 'u' and 'v' in an interval [a, b]. a) Prove that f is continuous at each point of [a, b] b) Assume that f is integrable on [a, b]. Prove that: \left | \int_{a}^{b}...

Homework Statement Multiple Choice If f is a continuous, decreasing function on [0, 10] with a critical point at (4, 2), which of the following statements must be false? E (A) f (10) is an absolute minimum of f on [0, 10]. (B) f (4) is neither a relative maximum nor a relative minimum...

hi guys, I have a question I would like assistance with: let (v,||.||) be a norm space over ℝ, and let f:v→ℝ be a linear functional. if f is continuous on 0 (by the metric induced by the norm), prove that there is k>0 such that for each u in v, |f(u)| ≤ k*||u||. thanks :)

People have found ways to extend the definition of some operations that are ostensibly discrete (such as differentiation - e.g. 1st, 2nd, 3rd derivatives) to operations that are defined for fractions ( e.g. fractional derivatives). Is there an interesting way to extend the operation of...

Homework Statement Let f be a continuous real function on a metric space X. Let Z(f) be the set of all p in X at which f(p) = 0. Prove that Z(f) is closed. Homework Equations Definition of continuity on a metric space. The Attempt at a Solution Proof. We'll show that X/Z(f) = {p...

Hi, All: I saw this question somewhere else: we are given any two topological spaces (X,T), (X',T'), and we want to see if there is always at least one continuous map between the two. The idea to say yes is this: we only need to find f so that f-1(U)=V , for every U in T', and some V in T. So...