Uniform continuity Definition and 82 Threads

Hello, Homework Statement Given that f is continuous in [1,\infty) and lim_{x->\infty}f(x) exists and is finite, prove that f is uniformly continuous in [1,\infty) The Attempt at a Solution We will mark lim_{x->\infty}f(x) = L . So we know that there exists x_{0} such that for...

Recently, I proved that Given f:A \rightarrow \mathbb R is uniformly continuous and (x_{n}) \subseteq A is a Cauchy Sequence, then f(x_{n}) is a Cauchy sequence, which really isn't too difficult a proof, however I'm having issues with the converse statement... More specifically, Suppose A...

I spent at least 2-3 hours thinking about this "deceivingly" (at least to me) simple problem but I just don't know how to proceed. Any hints would be greatly appreciated! Homework Statement Directly from the \varepsilon - \delta definition of uniform continuity, prove that 2^x is uniformly...

Homework Statement 1. Consider the function f(x) = x^3. Prove that (a) it is not uniformly continuous on R, but that (b) it is uniformly continuous on any interval of the type [-a, a] 2. Suppose that f is uniformly continuous on a region S, and g is uniformly continuous on the region f(S)...

I'm having some trouble understanding the proof for uniform continuity. I'm using the book Introduction to Real Analysis by Bartle and Sherbert 3rd Edition, page 138, if anyone has access to it. The Theorem states: I understand the proof up to the part where it says it is clear that...

Hi, This may sound lame but I am not able to get the definition of uniform continuous functions past my head. by definition: A function f with domain D is called uniformly continuous on the domain D if for any eta > 0 there exists a delta > 0 such that: if s, t D and | s - t | < delta...

Homework Statement let f(x)= (x^2)/(1+x) for all x in [ifinity, 0) proof that f(x) is uniformly continuous. can anyone help me with this problem Homework Equations using the definition of a uniform continuous function The Attempt at a Solution i did long division to simplify the...

From my textbook, this is the proof given for a theorem stating that any function continuous in a closed interval is automatically uniformly continuous in that interval. Proof: "If f were not uniformly continuous in [a, b] there would exist a fixed \epsilon > 0 and points x, z in [a, b]...

Homework Statement Show that f(x)=\frac{1}{x^{2}} is uniformly continuous on the set [1,\infty) but not on the set (0,1]. Homework Equations The Attempt at a Solution I've been working at this for at least 2 hours now, possibly 3, and I can't say I really have much of any idea...

Homework Statement suppose f and g are uniformly continuous functions on X and f and g are bounded on X, show f*g is uniformly continuous. The Attempt at a Solution I know that if they are not bounded then they may not be uniformly continuous. ie x^2 and also if only one is bounded...

Homework Statement if f and g are 2 uniformly continuous functions on X --> R show that f+g is uniformly continuous on X The Attempt at a Solution I tried showing that f+g is Lipschitz because all Lipschitz functions are uniformly continuous. So i end up with d(x_1,x_2) <...

Homework Statement If f and g are uniformly continuous on X, give an example showing f*g may not be uniformly continuous. The Attempt at a Solution i think if the functions are unbounded the product will not uniformly continuous. Is there a specific example of this function..?

Homework Statement Let f: [0,1] -> R (R-real numbers) be a continuous non constant function such that f(0)=f(1)=0. Let g_n be the function: x-> f(x^n) for each x in [0,1]. I'm trying to show that g_n converges pointwise to the zero function but NOT uniformly to the zero function...

Homework Statement Show that if a function f:(0,1) --> lR is uniformly continuous, f is bounded. Homework Equations - The Attempt at a Solution Really don´t know. I started thinking about Weierstrass Thereom but I am not sure that it´s ok. Now I think that may be is something...

A function f:D\rightarrowR is called a Lipschitz function if there is some nonnegative number C such that absolute value(f(u)-f(v)) is less than or equal to C*absolute value(u-v) for all points u and v in D. Prove that if f:D\rightarrowR is a Lipschitz function, then it is uniformly...

[SOLVED] Uniform Continuity Homework Statement Let A \subset \mathbb{R}^n and let f: A \mapsto \mathbb{R}^m be uniformly continuous. Show that there exists a unique continuous function g: \bar{A} \mapsto \mathbb{R}^m such that g(x)=f(x) \ \forall \ x \in A . Homework Equations...

Hi guys. Final tomorrow and i had some last minute questions for proving/disproving a function is uniformly cont. Basically i want to know if the following proofs are acceptable Consider f(x)=1/x for x element (0, 2) = I Proof 1: f(x) does not converge uniformly on I. In order for...

In my analysis class we were posed the following question: Give an example of a uniformly continuous function f: (0,1) ---> R' such that f' exists on (0,1) and is unbounded. we came up with the example that f(x) = x*sin(1/x) if you interpret the question to mean f' is unbounded, not f...

It began with my trying to prove that a uniformly continuous function on a bounded subset of the line is bounded. I took the hard route cause I couldn't figure out how to do this directly. I prove that if a real function is uniformly continuous on a bounded set E then there exists a continuous...

Homework Statement 1) f is some function who has a bounded derivative in (a,b). In other words, there's some M>0 so that |f'(x)|<M for all x in (a,b). Prove that f is bounded in (a,b). 2) f has a bounded second derivative in (a,b), prove that f in uniformly continues in (a,b)...

Homework Statement Prove that if y>=x>=0: a) y^2 arctan y - x^2 arctan x >= (y^2 - x^2) arctan x b) \ | \ y^2 arctan y - x^2 arctan x \ | \ >= (y^2 - x^2) arctan x c) use (b) to prove that x^2 arctan(x) isn't UC in R. Homework Equations The Attempt at a Solution a) We...

Homework Statement True or false: 1)If f is bounded in R and is uniformly continues in every finate segment of R then it's uniformly continues for all R. 2)If f is continues and bounded in R then it's uniformly continues in R. Homework Equations The Attempt at a Solution 1) If...

Homework Statement Show that if f: S -> Rn is uniformly continuous and S is bounded, then f(S) is bounded. Homework Equations Uniformly continuous on S: for every e>0 there exists d>0 s.t. for every x,y in S, |x-y| < d implies |f(x) - f(y)| < e bounded: a set S in Rn is bounded if it is...

I'm working on a proof that is not a homework assignment - that's why I'm posting it here. My question is simple. The epsilon-delta definition of continuity at a point a in an open subset E of the Complex plane is: \forall a \in E, \ \forall \ \varepsilon > 0 \ , \exists \ \delta > 0 \...

"Let f:R->R be differentiable such that |f'|<= 15, show that f is uniformly continuous." I can't solve it. I tried writing down the definition, but it got no where.

Prove that if f is uniformly continuous on a bounded set S then f is bounded on S. Our book says uniform continuity on an interval implies regular continuity on the interval, and in the previous chapter we proved that if a function is continuous on some closed interval then it is bounded...

"Let f:[a,b]\rightarrow [a,b] be defined such that |f(x)-f(y)|\leq a|x-y| where 0<a<1. Prove that f is uniformly continuous and (other stuff)." Let e>0 and let d=e/a. Whenever 0<|x-y|<d, |f(x)-f(y)|\leq a|x-y|<ad=e. f is therefore by definition uniformly continuous. Did I do this right? It...

"Suppose f:[0, inf) -> R is such that f is uniformly continuous on [a, inf) for some a>0. Prove that f is uniformly continuous on [0, inf)." But this is not true, is it? Consider the function f(x)=\left\{\begin{array}{cc}x &\mbox{ if }x\geq 1\\ \frac{1}{x-1} &\mbox{ if }x<1\end{array}\right

hello all i have been working on this problem, see i can see how it could be true but i don't know how to prove it,would anybody have any ideas on how to prove this ? let [a,b] be a closed interval in R and f:[a,b]->R be a continuous function. prove that f is uniformly continuous...

I'm seeking a bit of affirmation or correction here before i try to solidify this to memory... I know continuity to mean: Let f:D -> R (D being an interval we know to be the domain, D) Let x_0 be a member of the domain, D. This implies that the function f is continuous at the point...

There are two proofs which I have attempted to work on that have beein somewhat trifling. The first of which is : prove that the function f: [0,infin) -> R defined by f(x) = 1/x is not uniformly continuous on (0,infin). im thinking that the way in which i should probably attempt to solve...

I am a little shaky with the concept of proving uniform continuity vs regular continuity. Is the difference when proving through epsilon-delta definition just that your delta can not depend on "a" (thus be defined in terms of "a") (when |x-a|<delta) for uniform continuity? Also to the more...