Uniform continuity Definition and 84 Threads

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.

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  1. estro

    Uniform continuity of composite function

    I'll be very thankful is someone will tell me where I'm wrong. We know: 1) f is uniform continuous. 2) g is uniform continuous. We want to prove: fg(x) is uniform continuous. proof: from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e from 2 we know -> for every |x-y|<d...
  2. S

    Uniform Continuity proof, does it look reasonable?

    Homework Statement Note: I will use 'e' to denote epsilon and 'd' to denote delta. Using only the e-d definition of continuity, prove that the function f(x) = x/(x+1) is uniformly continuous on [0, infinity). Homework Equations The Attempt at a Solution Proof: Must show...
  3. T

    Proving Uniform Continuity of f on [1, $\infty$]

    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...
  4. C

    Converse Statement of Uniform Continuity

    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...
  5. F

    Uniform continuity of 2^x over [0,n]

    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...
  6. T

    Proving Uniform Continuity for Composite Functions

    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)...
  7. D

    Why Is Uniform Continuity Proven by Contradiction?

    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...
  8. S

    Uniform Continuity: Definition & Applications

    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...
  9. M

    Proving Uniform Continuity of f(x): Let x in [Infinity, 0)

    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...
  10. J

    Uniform Continuity Proof for Functions on Closed Intervals

    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]...
  11. L

    Uniform Continuity of 1/x^2 on various sets

    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...
  12. C

    Uniform Continuity: Showing f*g Is Uniformly Continuous on Bounded X

    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...
  13. C

    Proving Uniform Continuity for f+g on X

    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) <...
  14. C

    Uniform Continuity: Example of f*g Not Being Uniformly Continuous

    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..?
  15. C

    Show Pointwise Convergence of g_n to Zero 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...
  16. 3

    Is Uniform Continuity a Property of Bounded Functions?

    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...
  17. A

    Lipschitz function and uniform continuity

    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...
  18. V

    Solved: Proving Uniform Continuity of Unique Continuous Function on A

    [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...
  19. S

    Uniform Continuity- extentions of functions

    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...
  20. M

    Uniform continuity and boundedness

    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...
  21. S

    Characterization of Uniform Continuity on the line

    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...
  22. daniel_i_l

    Uniform continuity and derivatives

    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)...
  23. daniel_i_l

    Prove Uniform Continuity: y^2 arctan y - x^2 arctan x

    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...
  24. daniel_i_l

    Uniform continuity with bounded functions

    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...
  25. R

    Uniform continuity, bounded subsets

    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...
  26. B

    Proof Uniform Continuity: Epsilon-Delta & Sequential Equivalents

    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 \...
  27. T

    Show Uniform Continuity: Let f:R->R be Differentiable with |f'|<=15

    "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.
  28. 1

    Uniform continuity and bounded

    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...
  29. I

    Proving Uniform Continuity for Functions with a<1

    "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...
  30. I

    Is f(x) uniformly continuous on [0, inf)?

    "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
  31. S

    How Can You Prove Uniform Continuity on a Closed Interval?

    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...
  32. M

    Are My Definitions of Continuity and Uniform Continuity Correct?

    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...
  33. M

    Calc Proof(s): Uniform Continuity

    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...
  34. Z

    Uniform Continuity (and Irrationals)

    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...
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