Continuous functions Definition and 133 Threads

Let f be a uniformly continuous function on Q... Prove that there is a continuous function g on R extending f (that is, g(x) = f(x), for all x∈Q I think I am supposed to somehow use the denseness of Q and the continuity of a function to prove this, but I am not quite sure where I should start...

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...

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 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...

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

Let I be the interval I=[0,infinity). Let f: I to R be uniformly continuous. Show there exist positive constants A and B such that |f(x)|<=Ax+B for all x that belongs to I. Please help me!~

Homework Statement assume h: R->R is continuous on R and let K={x: h(x)=0}. Show that K is a closed set. Homework Equations The Attempt at a Solution since we know h is continuous and h(x)=0. therefore, we know there is a epsilon neighborhood such that x belongs to preimage...

Homework Statement Let Ce([0,1], R) be the set of even functions in C([0,1], R), show that Ce is closed and not dense in C. Homework Equations The Attempt at a Solution I think I can solve this if I can show that even functions converge to even functions, but I can't quite...

Hi, I think I've gotten this problem, but I was wondering if somebody could check my work. I still question myself. Maybe I shouldn't, but it feels better to get a nod from others. Homework Statement Consider the space of functions C[0,1] with distance defined as...

Homework Statement A mapping f from a metric space X to another metric space Y is continuous if and only if f^{-1}(V) is closed (open) for every closed (open) V in Y. Use this and the metric space (X,d), where X=C[0,1] (continuous functions on the interval [0,1]) with the metric d(f,g)=\sup...

This is a question from Papa Rudin Chapter 2: Find continuous functions f_{n} : [0,1] -> [0,\infty) such that f_{n} (x) -> 0 for all x \in [0 ,1] as $n -> \infty. \int^{1}_{0} f_n dx -> 0 , but \int^{1}_{0} sup f_{n} dx = \infty. Any idea? :) Thank you so much!

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?)

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 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?

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 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...

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...

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...

This is a simple question: given a continuous function, f(x), in a closed interval, how do we show that there is value "a" small enough such as for arbitrary x: f(x+a) - f(x) < e Where e lower bound is 0. ? Thxs in advance.

Let f: D \rightarrow \mathbb{R} be continuous. Is there an easier function that counterexamples; if D is closed, then f(D) is closed than D={2n pi + 1/n: n in N}, f(x)=sin(x) ? Plus, these counterexamples are all the same with the domain changed, just correct me if I'm wrong. If...

The question reads: Is it true that every compact subset of \mathbb{R} is the support of a continuous function? If not, can you describe the class of all compact sets in \mathbb{R} which are supports of continuous functions? Is your description valid in other topological spaces? The answer to...

"Let h and g be uniformly continuous on I\subset\mathbb{R} and are both bounded. Show that hg is uniformly continuous." h and g are uniformly continuous and bounded on I implies that h and g are continuous and bounded on I. This implies that hg (h times g) is continuous and bounded on I. Let...

hi, My question reads: Let f be defined and continuous on the interval D_1 = (0, 1), and g be defined and continuous on the interval D_2 = (1, 2). Define F(x) on the set D=D_1 \cup D_2 =(0, 2) \backslash \{1\} by the formula: F(x)=f(x), x\in (0, 1) F(x)=g(x), x\in (1, 2)...

I need to find a value for f at (0,0) to make this function continuous: f(x,y)=sqrt(x^2+y^2)/[abs(x) + abs(y)^(1/3)] With other functions in this problem I simply took the limit .. but taking the limit gives 0/0. In single-variable calculus I would apply l'hopital's rule to this, but I'm...

I would like a proof or a counter-example for the following claim: A non-constant real-valued continuous function (f:R->R) cannot have an arbitirarly small period!

There are several definitions of a continuous function between metric spaces. Let (X,d_X) and (Y,d_Y) be metric spaces and let f:X\rightarrow Y be a function. Then we have the following as definitions for continuity of f: \square \quad \forall\, x \in X \mbox{ and } \forall \, \epsilon >0 \...

Let D[a,b] be the set of piece-wise continuous functions on the close interval [a,b]. Show that D[a,b] is a subspace of the vector space P[a,b] of all functions defined on the interval [a,b]. Can someone get me started? Do I just need to show that they are closed under addition and...

Cb is set of all complex valued bounded functions R is set of Real numbers Define F:Cb(R)->Cb(R) by F(f)=f^2 for all 'points' f is an element of Cb(R). Prove that F is continuous. Can someone give me some guidance on how to get started with this one?

lo, I've got a quick q about the equation in the title, I've been asked to show/prove by analysis, that if f and g are continuous functions then M(x) is also continuous, it seems pretty intuitive but i just don't know how they want us to prove it, any help would be gr8

I want to determine all continuous functions s.t. for all x, y reals: f(x+y)f(x-y) = {f(x)f(y)}^2 Now, I want to know where on the web I can learn how I go about doing this, because I don't know what methods to use or what I should be aiming at in manipulating the above. I don't want...