What is Uniform continuity: Definition and 84 Discussions
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
Continuous functions can fail to be uniformly continuous if they are unbounded on a finite domain, such as
f
(
x
)
=
1
x
{\displaystyle f(x)={\tfrac {1}{x}}}
on (0,1), or if their slopes become unbounded on an infinite domain, such as
f
(
x
)
=
x
2
{\displaystyle f(x)=x^{2}}
on the real line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).
Although ordinary continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.
This is a bit of a longer post. I have tried to be as brief as possible while still being self-contained. My questions probably do not have much to do with ODEs, but this is the context in which they arose. Grateful for any help.
In what follows ##|\cdot|## denotes either the absolute value of...
I'd say yes, it is. Suppose ##|f|## is uniformly continuous on ##D##.
Then for all ##\epsilon>0## there is ##\delta>0## (call this ##\delta'##) such that if ##x,y\in D##, then ##||f(x)|-|f(y)||<\epsilon##.
Define sets:
##D^+=\{x\in D: x>a\}##
##D^-=\{x\in D: x<a\}##
Restrict the domain of...
There are two parts to the question Let's start with part :)
I understand the definition of Uniform continuity And I think I'm in the right direction for the solution but I'm not sure of the formal wording.
So be it ε>0
Given that yn limyn-xn=0 so For all ε>0 , ∃N∈ℕ so that For all N<n ...
I came across the following question:
If g and f are uniform continuity functions In section I, then f + g uniform continuity In section I.
I was able to prove it with the help Triangle Inequality .
But I thought what would happen if they asked the same question for f-g
I'm sorry if my...
In my textbook when proving continuity implies uniform continuity (which is very similar to the proof given here), BWT is used to find a converging subsequence. I cannot see why this is needed. Referring to the linked proof, if we open up the inequality ##|x_n-y_n|<\frac{1}{n}##, isn't by the...
I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
Homework Statement
Show that ##f(x)=\frac{1}{x^2}## is not uniformly continuous at ##(0,\infty)##.
Homework Equations
N/A
The Attempt at a Solution
Given ##\epsilon=1##. We want to show that we can compute for ##x## and ##y## such that ##\vert x-y\vert\lt\delta## and at the same time ##\vert...
I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ...
I am focused on Chapter 1: Continuity ... ...
I need help with an aspect of the proof of Theorem 1.8.15 ... ...
Duistermaat and Kolk's Theorem 1.8.15 and its proof read as follows:In...
I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...
I am focused on Chapter 5: Continuous Functions ...
I need help in fully understanding an aspect of Example 5.4.6 (b) ...Example 5.4.6 (b) ... ... reads as follows:
In the above text...
Homework Statement
Prove that f(x) = 1/(|x|+1) is uniformly continuous on R.
Homework EquationsThe Attempt at a Solution
This needs to be an e-d proof (epsilon-delta).
So I suppose we should start with let e>0, then we want to find a d such that for all x,y in R, if |x-y|<d then...
Homework Statement
Let ##f:X \to Y##. Show that
##f## not uniform continuous on ##X## ##\Longleftrightarrow## ##\exists \epsilon > 0## and sequences ##(p_n), (q_n)## in ##X## so that ##d_X(p_n,q_n)\to 0 ## while ##d_Y(f(p_n),f(q_n))\ge \epsilon##.
Homework Equations
Let ##f:X\to Y##. We say...
Let us have a continuous function f which is uniformly continuous on [a,b] and [b,c]...
Then Spivak says, f is uniformly continuous on [a,c]...
For prving this, he invokes the continuity of f on b...
My questions here are:
1.For a given ε, we have a δ1 which works on whole of interval [a,b] and...
I would appreciate it if someone could explain the steps in the reasoning of the following statement. This is not a homework assignment or anything.
Let ##U \subseteq R^{n}## be compact and ##f:U\to R## a continuous function on ##U##. However f is not uniformly continuous.
Then there exists an...
I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be...
Hello,
I've been attempting to do these problems from my textbook:
1. Suppose that f is a continuous function on a bounded set S. Prove that the
following two conditions are equivalent:
(a) The function f is uniformly continuous on S.
(b) It is possible to extend f to a continuous function on...
Let a\in\mathbb{R}, a>0 be fixed. We define a mapping
\mathbb{Q}\to\mathbb{R},\quad q\mapsto a^q
by setting a^q=\sqrt[m]{a^n}, where q=\frac{n}{m}. How do you prove that the mapping is locally uniformly continuous? Considering that we already know what q\mapsto a^q looks like, we can define...
Let $f:[0,\infty)\to\mathbb R$ be a differentiable function such that for all $a>0$ exists a constant $M_a$ such that $|f'(t)|\le M_a$ for all $t\in[0,a]$ and $f(t)\xrightarrow[n\to\infty]{}0.$ Show that $f$ is uniformly continuous.
Basically, I need to prove that $f$ is uniformly continuous...
Suppose I have a function f(x) \in C_0^\infty(\mathbb R), the real-valued, infinitely differentiable functions with compact support. Here are a few questions:
(1) The function f is trivially uniformly continuous on its support, but is it necessarily uniformly continuous on \mathbb R?
(2) I...
I'm struggling with the concept of uniform continuity. I understand the definition of uniform continuity and the difference between uniform and ordinary continuity, but sometimes I confuse the use of quantifiers for the two.
The other problem that I have is that intuitively I don't...
What is the difference between Lipschitz continuous and uniformly continuous? I know there different definitions but what different properties of a function make them one or the other(or both).
So Lipschitz continuity means the functions derivative(gradient) is bounded by some real number and...
hello
(pardon me if this is a lame question, but i got to still ask)
If a function is uniformly continuous (on a given interval) then is it required for the derivative of the function to be continuous?
I was thinking as per the definition of Uniform continuity, f(x) should be as close to...
Homework Statement
Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S.Homework Equations
Uniform continuity: For all e>0, there exist d>0 s.t for all x,y in S |x-y| implies |f(x)-f(y)|
The Attempt at a Solution
Every time my book has covered a...
Homework Statement
For question 19.2 in this link:
http://people.ischool.berkeley.edu/~johnsonb/Welcome_files/104/104hw7sum06.pdf
I came up with a different proof, but I'm not sure if it is correct...
Homework Equations
The Attempt at a Solution
Let |x-y|< \delta
For...
I'm taking my first course in Analysis, and we learned a couple of theorems about Uniform Continuity. I have been able to visualize most of what's been going on before, but I need some help with the following:
E \subseteq ℝ, f: E \rightarrow ℝ uniform continuous. if a sequence xn is Cauchy...
I don't understand uniform continuity :(
I don't understand what uniform continuity means precisely. I mean by definition it seems that in uniform continuity once they give me an epsilon, I could always find a good delta that it works for any point in the interval, but I don't understand the...
Homework Statement
(X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact,
g(y) is a continuous function that maps Y->Z with a continuous inverse
If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x))
Show that if h is uniformly continuous, f is uniformly...
Homework Statement
Prove that if F is uniformaly continuous on a bounded subset of ℝ, then F is bounded on A.
Homework Equations
The Attempt at a Solution
F is uniformaly continuous on a bounded subset on A in ℝ. Therefore each ε>0, there exists δ(ε)>0 st. if x, u is in A where...
If the function f:D→ℝ is uniformly continuous and a is any number, show that the function a*f:D→ℝ also is uniformly continuous.
Ok, so I am just learning my proofs so be patient with me, I'm very new at it.
take a>0, ε>0 and x,y in D. We know |x-y|<δ whenever |f(x)-f(y)|<ε.
If we take...
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...
Homework Statement
This isn't really a problem but it is just something I am curious about, I found a theorem stating that you have two metric spaces and f:X --> Y is uniform continuous and (xn) is a cauchy sequence in X then f(xn) is a cauchy sequence in Y.
Homework Equations
This...
I'm working on a problem for my analysis class. Here it is:
Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S.
I'm not too sure that this question is...
Why some functions that are continuous on each closed interval of real line fails to be uniformly continuous on real line. For example x2. Give conceptual reasons.
This isn't so much of a homework problem as a general question that will help me with my homework.
I am supposed to prove that a given function is uniformly continuous on an open interval (a,b).
Since for any continuous function on a closed interval is uniformly continuous, I am curious...
Question: Show that f(x)= (x^2)/((x^2)+1) is continuous on [0,infinity). Is it uniformly continuous?
My attempt: So I know that continuity is defined as
"given any Epsilon, and for all x contained in A, there exists delta >0 such that if y is contained in A and abs(y-x)<delta, then...
Hi, All:
Let f R-->R be differentiable. If |f'(x)|<M< oo, then f is uniformly continuous, e.g.,
by the MVTheorem. Is this conditions necessary too, i.e., if f:R-->R is differentiable
and uniformly continuous, does it follow that |f'(x)|<M<oo ?
Thanks.
Homework Statement
1)Show, if E is a subset of D is a subset of the real numbers R and f maps D into R is uniformly continuous, then the restriction of f to E is also uniformly continuous.
2)Show, if f is continuous and real valued on [a,b) and if the limit of f(x) as x approaches b...
determine if these functions are uniformly continuous ::
1- \ln x on the interval (0,1)
2- \cos \ln x on the interval (0,1)
3- x arctan x on the interval (-infinty,infinty)
4- x^{2}\arctan x on the interval (infinty,0
5- \frac{x}{x-1}-\frac{1}{\ln x} on the interval (0,1)...
Homework Statement
If f:S->Rm is uniformly continuous on S, and {xk} is Cauchy in S show that {f(xk)} is also cauchy.
Homework Equations
The Attempt at a Solution
Since f is uniformly continuous,
\forall\epsilon>0, \exists\delta>0: \forallx, y ∈ S, |x-y| < \delta =>...
Homework Statement
Prove that f:(M,d) -> (N,p) is uniformly continuous if and only if p(f(xn), f(yn)) -> 0 for any pair of sequences (xn) and (yn) in M satisfying d(xn, yn) -> 0.
Homework Equations
The Attempt at a Solution
First, let f:(M,d)->(N,p) be uniformly continuous...
[PLAIN]http://img258.imageshack.us/img258/78/52649134.jpg
So I've thought of a few ideas on how to prove this, but only one so far that I've sort of figured out what to do. What I want to do is split the interval up in two, so from [0,b] and from (b, ∞), for some b in the reals. Now since f is...
I was wondering if f and g are two uniformly continuous functions on a set such that g(x) is not zero is f/g uniformly continuous?
I have a feeling it is not but I can't seem to find a counter example.
Homework Statement
Assume f:(0,1) \rightarrow \mathbb{R} is uniformly continuous. Show that \lim_{x \to 0^+}f(x) exists.Homework Equations
Basic theorems from analysis.The Attempt at a Solution
The statement is intuitive but I'm having trouble formalizing the idea. Uniform Continuity means...
Homework Statement
a) Give an example of a bounded continuous function f: R -> R which is not uniformly continuous.
b) State (in terms of a small Epsilon and a large K) what it means to say that f(x) -> 0 as x -> infinity (plus or minus)
c) Now assume that f: R -> R is continuous and...
Homework Statement
Suppose that f: [0, \infty) \rightarrow \mathbb{R} is continuous and that there is an L \in \mathbb{R} such that f(x) \rightarrow L as x \rightarrow \infty. Prove that f is uniformly continuous on [0,\infty).
2. Relevant theorems
If f:I \rightarrow \mathbb{R} is...
hi everyone
I was reading one example about Uniform continuity, say that the polynomials, of degree less than or equal that 1 are Uniform continuity, my question is, for example in the case polynomial of degree equal to one Which is \delta, that the Uniform continuity condition satisfies...
Homework Statement
Show that if h is continuous on [0, ∞) and uniformly continuous on [a, ∞),
for some positive constant a, then h is uniformly continuous on [0, ∞).
Homework Equations
The Attempt at a Solution
I'm thinking of using the epsilon-delta definition of continuity...
So my teacher said that uniform continuity was a metric space notion, not a topological space one. At first it seemed obvious, since there is no "distance" function in general topological spaces. But then I remembered that you can generalize point-wise continuity in general topologies, so why...