Continuous functions Definition and 133 Threads

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 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...

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 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...

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...

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

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 :)

Homework Statement Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero. Homework Equations A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')}...

Homework Statement Suppose that f(x) is a continuous function on [0,2] with f(0) = f(2). Show that there is a value of x in [0,1] such that f(x) = f(x+1). Homework Equations Intermediate Value Theorem? Extreme Value Theorem? Periodicity? The Attempt at a Solution For sure there's an...

Homework Statement Let f : R → R be continuous on R and assume that P = {x ∈ R : f(x) > 0} is non-empty. Prove that for any x0 ∈ P there exists a neighborhood Vδ(x0) ⊆ P. Homework Equations The Attempt at a Solution If you choose some x, y ∈ P, since f(x) is continuous then |f(x)...

I did the work. I'm not sure on some of these. I think for (c) I need to make D = (0, infinity) http://i111.photobucket.com/albums/n149/camarolt4z28/1-3.png http://i111.photobucket.com/albums/n149/camarolt4z28/2-3.png http://i111.photobucket.com/albums/n149/camarolt4z28/3-1.png

I suppose my question is, "does the set of all continuous functions comprise a continuum?" How would one even start at trying to prove that? Any ideas or suggestions?

Homework Statement Define the function at a so as to make it continuous at a. f(x)=\frac{4-x}{2-\sqrt{x}}; a = 4 Homework Equations \lim_{x \rightarrow 4} \frac{4-x}{2-\sqrt{x}} The Attempt at a Solution I cannot think of how to manipulate the denominator to achieve f(4), so I...

Could you please check the statement of the theorem and the proof? If the proof is more or less correct, can it be improved? Theorem Let be a topological space and be the discrete space. The space is connected if and only if for any continuous functions , the function is not onto...

I was reading a Ph.D. thesis this morning and came across the claim that "a uniform limit of absolutely continuous functions is absolutely continuous." Is this true? What about the sequence of functions that converges to the Cantor function on [0,1]? Each of those functions is absolutely...

Homework Statement Find an example of a continuous function f:R->R with the following property. For every epsilon >0 there exists a delta >0 such that |f(x)-f(y)| <epsilon whenever x,y e R with |x-y|<delta. Now find an example of a continuous function f:R->R for which this property does nto...

I was studying particle in a box from shankar and I couldn't get the following point. If V is infinite at for x > L/2 and x < L/2, so is double derivative of psi. Now Shankar mentions that it follows the derivative of psi has a finite jump. I am not able to get this point because according to my...

If f is a continuous functional on a normed space, do you have \sup_{\|x\| < 1} |f(x)| = \sup_{\|x\| = 1} |f(x)| If so, why? If not, can someone provide a counterexample?

I'm trying to understand the set C_0(X), defined here as the set of continuous functions f:X\rightarrow\mathbb C such that for each \varepsilon>0, \{x\in X|\,|f(x)|\geq\varepsilon\} is compact. (If you're having trouble viewing page 65, try replacing the .se in the URL with your country domain)...

Question Let (S; d) and (T;D) be metric spaces. A function f : X -> Y is said to be uniformly continuous if ( for all epsilons > 0)(there exists a sigma > 0) such that d(x; y) < sigma => D(f(x); f(y)) < epsilon a. Show that a uniformly continuous function maps Cauchy sequences to Cauchy...

Homework Statement Let (X,T) be a normal topological space. Let R be the ring of continuous real-valued functions (with respect to the given topology T) from X onto the real line. Prove that the that T is the coarsest Topology such that every function in R is continuous. Homework...

Homework Statement Let f: R-->R be continuous. For δ>0, define g: R-->R by: g(x) = (1/2δ) ∫ (from x-δ to x+δ) f Show: a) g is continuously differentiable b) If f is uniformly continuous, then, for every ε>0, there exists a δ1>0 such that sup{∣f(x) - g(x)∣; x∈R} < ε for 0<δ≤δ1The Attempt at...

Homework Statement Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space? Homework Equations None. The Attempt at a Solution I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g...

Homework Statement give an example of a function f: R --> R that is differentiable n times at 0, and discontinous everywhere else. Homework Equations ---The Attempt at a Solution i got one, and i proved everything, i just want to make sure what i did is correct: f:x n+1 when x is rational...

Homework Statement As in the question - Suppose that f_n:[0,1] -> Reals is a sequence of continuous functions tending pointwise to 0. Must there be an interval on which f_n -> 0 uniformly? I have considered using the Weierstrass approximation theorem here, which states that we can find...

I like thinking of practical examples of things that I learn in my analysis course. I have been thinking about functions fn:[0,1] --->R. What is an example of a sequence of piecewise functions fn, that converge uniformly to a function f, which is not piecewise continuous? I've thought of...

Homework Statement Let E,E' be metric spaces, f:E\rightarrow E' a function, and suppose that S_1,S_2 are closed subsets of E such that E = S_1 \cup S_2. Show that if the restrictions of f to S_1,S_2 are continuous, then f is continuous. Also, if the restriction that S_1,S_2 are closed is...

Here's the problem: Let f(x)={x, x in Q; 0, x in R\Q. Show f is continuous at c if and only if c = 0. Hint: You may want to use the following theorem: Let A and B be two disjoint subsets of R and f1:A\rightarrowR and f2:B\rightarrowR. Define f:A\cupB\rightarrowR by f(x)={f1(x), x in A...

Homework Statement give an example of functions f and g, both continuous at x=0, for which the composite f(g(x)) is discontinuous at x=0. Does this contradict the sandwich theorem? Give reasons for your answer. Homework Equations The Attempt at a Solution I understand the...

Is the space of all absolutely continuous functions complete? I've never learned about absolutely continuous functions, and so I'm unsure of their properties when working with them. I'm fairly certain it is, but would like some verification. Or a link to something on them besides the...

I'm struggling to understand continuous functions as subspaces of each other. I use ⊆ to mean subspace below, is this the correct notation? I also tried to write some symbols in superscript but couldn't manage. Anyway I know that; Pn ⊆ C∞(-∞,∞) ⊆ Cm(-∞,∞) ⊆ C1(-∞,∞) ⊆ C(-∞,∞) ⊆ F(-∞,∞) I...

We know that dim(C[a,b]) is infinte. Indeed it cannot be finite since it contains the set of all polynomials. Is the dimension of a Hamel basis for it countable or uncountable? I guess if we put a norm on it to make a Banach space, we could use Baire's to imply uncountable. I am however...

Homework Statement This is the last part of a revision question I'm trying, would really like to get to the end so any pointers or help would be greatly appreciated. Suppose h:(0,1)-> satisfies the following conditions: for all xЭ(0,1) there exists d>0 s.t. for all x'Э(x, x+d)n(0,1) we...

Continuous Functions - Setting up word problems Homework Statement Each side of a square is expanding at 5 cm/sec. What is the rate of change when the length of the sides are 10 cm. Homework Equations A = ab The Attempt at a Solution a = 5t, b = 5t and the area is...

Homework Statement Let f, g be continuous from R to R (the reals), and suppose that f(r) = g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x \in R?Homework Equations The Attempt at a Solution Basically, this seems trivial, but is probably tricky after all. I know that...

Hi guy's I know this is more of a homework question, I posted a similar thread earlier on but I think I ended up confusing myself. I need to show that a function is continuous between metric spaces. I'll post the question and what I've done any tips on moving forward would be great. I...

hints? Derivatives: Intervals, stationary points, logarithms, continuous functions Homework Statement Got any hints or anything? 1. Suppose that f(x) = (x - 3)^4 ( 2x + 5)^5 a) Find and simplify f ' ( x ) b) Find stationary points of f c) Find exactly the intervals where f is...

Homework Statement What is the cardinality of the set of all continuous real valued functions [0,1] \rightarrow R . The Attempt at a Solution In words: I will be using the Cantor Bernstien theorem. First the above set, let's call it A, is lesser then or equal to the set of all...

Homework Statement g(x)={x+3, x=3 {2+\sqrt{k} , x=3 find k if g(x) is continuous Homework Equations The Attempt at a Solution I have no idea how to begin, but drawing the first part on a cartesian plane.

I hope someone can help me wif this qnestion. Qn : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is x0 ∈ [0, 1] such that f (g(x0)) = g(f (x0)). Thanks in advance.

Hi all, The correlation coefficients (Pearson's) is usually defined in terms of discrete sampling of a function. However, I have seen that the mean and standard deviation, for example, are also typically written in terms of discrete variables BUT may also be expressed in terms of a...

Homework Statement Suppose f: [0,1] \rightarrow R is two-to-one. That is, for each y \in R, f^{-1}({y}) is empty or contains exactly two points. Prove that no such function can be continuous. Homework Equations Definition of a continuous function: Suppose E \subset R and f: E...

Homework Statement Prove the set of continuous functions from R to R has the same cardinality as RHomework Equations We haven't done anything with cardinal numbers (and we won't), so my only tools are the definition of cardinality and the Schroeder-Bernstein theorem and its consequences. I...

If f and g are continuous functions with f(4)-5 and lim [5f(x) -g(x) ]=5 find g(4) x-->4

I am confused with sequence and continuous functions. I am confued with their limit. how do they know the min and max before they attempt the question. and is that the only solution to the question? I mean. Everytimes if I see kind question like this, is that only way to do it?... Many...

[b]1. Suppose that f and g are continuous functions defined on R and every interval (a, b) contains some point y with f(y) = g(y). Show that f(x) = g(x) for every x in R. [b]3. I can show that between any two points in are there is some x such that f(x)=g(x). Is that enough? I don't think...

I was just reflecting upon my math courses and wondered why can we transform any piecewise continuous functions by using transforms such as laplace transforms or converting to Fourier series by simply adding the required integrals on the respective bounds?

Ok, so this was assigned as a bonus problem in my Topology class a while ago. Nobody in the class got it, but I've still been racking my brain on it ever since. ____ For some n, consider the set of all nxn nonsingular matrices, and using the usual Euclidean topology on this space, show that...