Continuous Definition and 1000 Threads

Can anyone give me an example of a function f:[a,b]->R which is continuous almost everywhere yet unbounded? Thanks!

Homework Statement If f and g are continuous functions, with f(3) = 5 and \stackrel{lim}{x\rightarrow3}\left[2f(x) - g(x)\right] = 4 find g(3) The Attempt at a Solution I'm stumped! I cannot find anything in my notes on where to begin. I am not looking for a specific answer, I just need...

Let F: R x R -> R be defined by the equation F(x x y) = { xy/(x^2 + y^2) if x x y \neq 0 x 0 ; 0 if x x y = 0 x 0 a. Show that F is continuous in each variable separately. b. Compute the function g: R-> R defined by g(x) = F(x x x) c. Show that F is not continuous. I know how to do...

Homework Statement Let C be the set of points where f: R --> R is continuous. Show that C may be written as the intersection of a countable collection of open sets in R. The attempt at a solution If C is empty, the result is true. If C has countably many points, say x_0, x_1, ..., then R...

Homework Statement Is the following uniformly continuous on (1,∞)? \int_{x}^{x^2}\: e^{-t^2}\mathrm{d}tHomework EquationsThe Attempt at a Solution Quite honestly I don't know where to start. I mean, I'm positive I have to use the theorem that says that if for all ε > 0 there exists δ > 0 so...

Homework Statement Let F be the set of all continuous functions with domain [-1,1] and codomain R. Let A be the algebra of all polynomials that contain only terms of even degree (A is a subset of F). Show that the closure of A in F is the set of even functions in F. The attempt at a...

Homework Statement f:[0,1] \rightarrow R is a continuous function such that \intf(t)dt (from 0 to x) = \int f(t)dt( from x to 1) for all x\in[0,1] . Describe f. Homework Equations integral represents area The Attempt at a Solution what ever the function is, I know that...

If u : R^2 \to R has continuous partial derivatives at a point (x_0,y_0) show that: u(x_0+\Delta x, y_0+\Delta y) = u_x(x_0,y_0) + u_y(x_0,y_0) + \epsilon_1 \Delta x + \epsilon_2 \Delta y, with \epsilon_1,\, \epsilon_2 \to 0 as \Delta x,\, \Delta y \to 0 I know this can be proved using MVT...

Regarding the definition of homemorphism, when we say a function is a homeomorphism if it is continuous, bijective, and has a continuous inverse I assume that means over the codomain only. For example if we have a map from f: R -> (0,1) does f inverse need to be continuous on (0,1) only?

Homework Statement Find c such that it makes f(x) continuous. Homework Equations f(x)=\begin{cases} 2x+c&x < -5\\ 3x^2&-5 \leq x < 0\\ cx^2&0 \leq x\\ \end{cases} The Attempt at a Solution I know that \lim_{x\to 5^-}3x^2 = 2x+c and \lim_{x\to...

Homework Statement Let A \subset X ; let f : A \rightarrow Y be continuous; let Y be Hausdorff. Show that if f may be extended to a continuous function g: \overline{A} \rightarrow Y, then g is uniquely determined by f. Homework Equations The Attempt at a Solution If f is a...

Homework Statement Say we have a sequence of functions {fn: [a,b] -> R} such that each fn is nondecreasing. Suppose that {fn} converges point-wise to f. Prove that if f is continuous, then {fn} converges uniformly to f. The attempt at a solution Fix an x in [a,b] and let e > 0. Then we can...

Homework Statement Decide whether the following functions are continuous at a=0 Homework Equations f(x)=x^2 if x<0 and f(x)=sinx if x>=0 The Attempt at a Solution I don't really understand where the 'a' comes in the functions? Some help/hints as to how to start off this question...

Homework Statement Let f, f1, f2, f3, ... be continuous real-valued functions on the compact metric space E, with f = lim fn. Prove that if fi ≤ fj whenever i ≤ j, then f1, f2, ... converges uniformly. The attempt at a solution I was trying to reverse engineer the proof, but I'm stuck...

Give an example of a function f:(0,1)-->Reals which is continuous at exactly the irrational points in (0,1). I think the function f such that f(x)=1/n if x is rational in (0,1) (x=m/n for some n not 0) and f(x)= 0 if x is irrational in (0,1) should work. I get the reason why f is continuous...

I'm currently at uni, but have difficulty doing problems involving continuous charge distributions. Say there's a charge distribution dl, dA or dV on a length surface or volume respectively at a distance R away from a point i know i must integrate over total length area or volume (depending on...

Homework Statement Let f and g be continuous functions defined on all of R. Prove that if f(a) \neq g(x) for some a \epsilon R , then there is a number \delta > 0 such that f(x) \neq g(x) whenever |x-a| < \delta. Homework Equations I would like to please check if my proof is...

I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient. Dirac rather liberally talks about observables that have a continuous range of...

Is there any equation/formula for continuous compound interest to which money is added (or substracted from) periodically? Or can one be derived? Thanks i.e. monthly interest rate is 50% and we add 1$ every month (:bugeye:) Initially: 1$ 1st Month: 1.6 + 1 = 2.6$ 2nd month: 4.3 + 1...

I'm trying to follow an argument in Pauling's Introduction to QM w/applications to chemistry. He is a bit hand-wavy at parts, and I wanted to see if anyone could clarify. So we start off with the time-dependent Schrodinger equation. He makes the assumption that \Psi, our wavefunction, can be...

Hello, i am trying to remember the name of the guy who created a water fountain that seemed to recycle its own water by using pressure or something in order to siphon it back up to the start. I can't remember the era this guy was from but i am pretty sure it was before electricity so probably...

Homework Statement f:(0,1]->R be a continuous function. Is it possible that f does not have an absolute min or max. Give counter examples Homework Equations The Attempt at a Solution Since f is partially bounded, if I break the interval down into smaller sub intervals, each will...

[b]1.Is it true that if f is continuous onto function on a closed interval then f(x) must also be a closed interval. How about the other way around. f is continuous and onto on a open bounded interval and f(x) is a closed interval Homework Equations f:[0,1]-->(0,1) f:(0,1)-->[0,1] The...

Homework Statement suppose f : [0,infinity) -> R is continuous, and there is an L in R, s.t. f(x) -> L, as x -> infinity. Prove that f is uniformly continuous on [0,infinity). Homework Equations limit at xo: |x-xo| < delta then |f(x) -L| < epsilon continuous |x-x0| < delta then |f(x)...

Why is it that every continuous function is a derivative? I know that not every derivative is continuous, I just don't know really know why we would know that every continuous function is a derivative. I think is has something to do with the integral, but I don't know how. Any help?

Let f be a real-valued function on R satisfying f(x+y)=f(x)+f(y) for all x,y in R. If f is continuous at some p in R, prove that f is continuous at every point of R. Proof: Suppose f(x) is continuous at p in R. Let p in R and e>0. Since f(x) is continuous at p we can say that for all e>0...

Homework Statement how could i prove that cos x= sum (n=1 to 00) [((-1)^n) * x^(2n)/((2n)!)] is continuous and differentiable at each x in R Homework Equations the Taylor Expansion of cosine is the given equation The Attempt at a Solution basically i need to prove that the...

F:R-->R, the fourth derivative of f is continuous for all x... Homework Statement Suppose f is a mapping from R to R and that the fourth derivative of f is continuous for every real number. If x is a local maximum of f and f"(x)=0 (the second derivative is zero at x), what must be true of the...

f:[a,infinity)-->R is continuous with f(x) > 0 for all x in [a,infinity),... Suppose "a" belongs to R, and f:[a,infinity)-->R is continuous with f(x) > 0 for all x in [a,infinity) and limf(x)=1 (as x goes to infinity). Prove that there exists r>0 such that f(x)>r for all x in [a,infinity).

Homework Statement Prove that the function: \frac{2x-1}{x^2+1}, x \in \mathbb{R} is continuous. Homework Equations Definition 1. The function y=f(x) satisfied by the set Df is continuous in the point x=a only if: 10 f(x) is defined in the point x=a i.e. a \in D_f 20 there is bound...

Homework Statement An astronomer is trying to estimate the surface temperature of a star with a radius of 5.0* 10^8m by modeling it as an ideal blackbody. The astronomer has measured the intensity of radiation due to the star at a distance of 2.5* 10^{13}m and found it to be equal to 0.055...

Homework Statement Suppose f:[a,b] \rightarrow [a,b] is continuous. Prove that there is at least one fixed point in [a,b] - that is, x such that f(x) = x. Homework Equations The Attempt at a Solution I was going to try something with the IVT, but then I realized I wasn't sure what...

I am trying to understand continuous and discontinuous. These are two assignments I have for a class. I am just looking for some feedback... Let A= {1/n : n is natural} Then, f(x)= (x , if x in A) (0 , if x not in A) This is discontinuous on A but continuous on A...

Homework Statement Let f map [a,b]-->R be a continuous nonegative function. Suppose Integral f(x)dx from a to b = 0 show that f = 0 on [a,b] The Attempt at a Solution Just not sure if this is good or not.. so the lower sum <= 0 = integral f(x) dx but the lower sum must be 0...

Homework Statement Suppose f: E--> R is cont at x0 and x0 is an element of F contained in E. Define g:F--->R by g(x)=f(x) for all x elemts of F. Prove g is continuous at x0. Show by example that the continuity of g at x0 need not imply the continuity of f at x0. Homework Equations...

Homework Statement f(0,1) ---> R by f(x) =1/x^(1/2) -((x+1)/x)^(1/2). Can one define f(0) to make f continuous at 0? Homework Equations lx-x0l<delta lf(x)-f(x0l<epsilon The Attempt at a Solution My thought is that the limit must equal f(0), but I'm unsure of how to get f(0)...

Homework Statement Is the function f(x) = x * sin(1/x) for x =/= 0 and f(0) = 0 unifoirmly continuous on R? Homework Equations The Attempt at a Solution dom(f) = (-inf, inf) x,y in R and |x-y| < d imply |f(x) - f(y)| < e |x - 0| < d imply |x * sin(1/x) - 0 | < e ?

Homework Statement Show that a continuous function such that for all c in the reals, the equation f(x) = c cannot have two solutions Homework Equations The Attempt at a Solution I was thinking along the lines of a contradiction or somehow using intermediate value theorem but it...

Is there a function f:R-->R that is continuous at π and discontinuous at all other numbers? Thx

differentiable and uniformly continuous?? Homework Statement Suppose f:(a,b) -> R is differentiable and | f'(x) | <= M for all x in (a,b). Prove f is uniformly continuous on (a,b). Homework Equations The definition of uniform continuity is: for any e there is a d s.t. | x- Y | < d...

Hi there! I got some difficulty understanding a question regarding electrons Electrons which are initially at rest are subjected to a continuous force of 2E-12 N along a length of 2 miles and reach very near the speed of light. a) Determine how much time is required to increase the...

Homework Statement Let I be an interval in R, and let f: I-->R be one-to-one, continuous function. Then prove that f^(-1):f(I)-->R is also continuous. Homework Equations The Attempt at a Solution I started a thread yesterday and had some responses but the proofs became quite...

Homework Statement Let I be an interval in the real line, and let f map I --> R be a one-to-one, continuous function. Then prove that f^(-1) maps f(I) --> R is also continuous The Attempt at a Solution I've started with the definition of continuity but I don't see where to go next.

Homework Statement With knowledge of the orthogonality conditions for eigenfunctions with discrete eigenvalues, determine the orthonormal set for eigenfunctions with continuous eigenvalues. Use the definition of completeness to show that | a(k) |^2 = 1. 2. The attempt at a solution The first...

Hello! Could anybody give me an idea about this proof? knowing f_{i}:X\rightarrowR i=1,2 to show whether f_{3}=min{f_{1},f_{2}} is continuous! Thanks in advance, Regards

Homework Statement Give an example of a function f : R -> R where f is continuous and bounded but not uniformly continuous. Homework Equations A function f : D -> R and R contains D, with Xo in D, and | X - Xo | < delta (X in D), implies | f(X) - f(Xo) | < epsilon. Then f is continuous...

Hi, I am having some problems understanding the degree of a continuous map g:circle --> circle I have found a definition in Munkres (pg 367) that I can't really understand (I'm an engineering student with little algebraic topology) and one in Lawson (pg 181), Topology:A Geometric approach...

I have an assignment question " let (X,d) be a metric space. a is element of X. Define a function f maps X -> R by f(x) = d(a,x). show f is continuous." I'm not sure what this function looks like. Is f(x) = sqrt(a^2+x^2) and if it is I need abs(x-a) < delta?? I'm confused.

What device (thing with at least one moving part) in constant motion, has given the longest service?

Homework Statement An ambulance travels back and forth, at a constant speed, along a road of length L. At a certain moment of time an accident occurs at a point uniformly distributed on the road. (That is, its distance from one of the fixed ends of the road is uniformly distributed over...