# Questions tagged [uniform-continuity]

For questions involving the concept of uniform continuity, that is, "the $\delta$ in the definition is independent of the considered point".

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When solving real-analysis' problems I like to represent the functions involved and think geometrically what is going on. Today I got the following exercise : Let $f \in \mathcal{C}^1(\mathbb{R},\... • 499 8 votes 0 answers 541 views ### A sequence of functions that is uniformly continuous, pointwise equicontinuous, but not uniformly equicontinuous when their domain is noncompact I'm trying to prove my sequence of functions$(f_n) = \frac{n}{n+1}\cos(x^2)$on (0,1) is pointwise equicontinuous, uniformly continuous, but not uniformly equicontinuous. But, I'm having a lot of ... • 257 5 votes 0 answers 94 views ### Does$\lim_{x\to\infty}|f(x+h)-f(x)|=0$implies$f$is uniformly continuous? Question: Let$f\in C([0,\infty))$. And$\forall \ h\in\mathbb{R}$, $$\lim_{x\to\infty}|f(x+h)-f(x)|=0.$$ Show that$f$is uniformly continuous on$[0,\infty)$. I have some idea about this ... 5 votes 0 answers 102 views ### Functions$f: \mathbb{R}^n \to \mathbb{R}$such that$|f(x) -f(y)| \le C \prod_{i=1}^n |x_i - y_i|^{\alpha_i}$The standard definition of a Holder continuous function between metric spaces$X,Y$is a function$f: X \to Y$such that there exist$C>0$and$0 < \alpha \le 1$such that $$d_Y(f(x),f(y)) \le ... • 7,900 5 votes 0 answers 383 views ### Baby Rudin Exercise 4.13 Alternate Proof Verification I would like to know if my proof of ex 4.13 is correct. Thanks! Exercise 4.13 in Rudin asks: Let E be a dense subset of a metric space X, and let f be a uniformly continuous real function ... • 839 4 votes 0 answers 80 views ### Groups where every continuous function is uniformly continuous Let G be a topological group, and (X, d) a metric space. The function f : G \to X is left uniformly continuous if for all \varepsilon > 0 there exists an identity neighbourhood U such ... • 2,855 4 votes 0 answers 83 views ### Conditional Expectation(E[f(X)\mid X \in S]) is uniformly continous in its argument(S) I basically want to prove that under some conditions E[f(X)\mid X \in S] is uniformly continuous in S To be more specific, suppose f:K \subset \mathbb{R}^n \rightarrow \mathbb{R} is a bounded ... • 81 4 votes 0 answers 55 views ### Uniform continuity of \frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x} Let f(x)=\frac{1}{2} \sin{ \frac{x}{2}}- \frac{1}{x} if 0<x \leqslant \pi and f(x)=0 if x=0. I took the sequences$$x_n=\frac{1}{n}y_n=\frac{1}{n^2}$$We have that x_n-y_n \... • 21.6k 4 votes 0 answers 463 views ### In what sense is metric space completion universal? The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ... • 20k 4 votes 0 answers 555 views ### Uniform Continuity This question has three parts. a) Difference between continuity and uniform continuity b) Geometrical meaning of uniform continuity c) Correct the example Definition of Continuity of a function in ... 4 votes 1 answer 136 views ### Is this an alternate way to check if a function is uniformly continuous? My question is whether the following statement is true, or if there exists one similar. For a differentiable real function f:S\rightarrow \mathbb{R} where S is an interval, and a fixed c \in S, ... • 2,419 4 votes 0 answers 98 views ### Find a constant C_p that satisfies |f(x+p)-f(y+p)|\le C_p|f(x)-f(y)| Let B^n be the unit open ball in \mathbb{R}^n, p\in \mathbb{R}^n and f\colon \mathbb{R}^n\to B^n defined as f(x)=\frac{1}{1+|x|}x. I believe there are constants C_p>0 such that |f(x+p)... 3 votes 1 answer 1k views ### problem about uniform continuity Problem: "Show that if f:\left[a,+\infty\right] \rightarrow \mathbb R is continuous and if \lim_{x \rightarrow + \infty} f(x) exists, then f is uniformly continuous". I have a question about ... • 1,077 3 votes 0 answers 88 views ### Example of a compact operator that is not uniformly continuous. [Solved] I want to find a Banach space E and a compact operator K:[0,1]\times E \rightarrow E (that is, K maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ... 3 votes 0 answers 63 views ### Proving f:[-1,1] \to \mathbb{R} with f(x)=x^3 is uniformly continuous Can anyone verify the steps in my proof are correct? Scratch work: |f(x)-f(y)| =|x^3-y^3|=|x-y||x^2+xy+y^2| (at this point note that we already know |x-y|<\delta for some \delta>0)$$|x-y||... 3 votes 1 answer 54 views ### Is$T$uniformly continous? Let$(X,\lVert \cdot\lVert)$be a normed space. Let$T$be a function given by \begin{equation} \begin{array}{cccc} T:&X&\longrightarrow&X\\ &x&\longmapsto&T(x)=x\lVert x\... 3 votes 0 answers 91 views ### (Uniform) convergence rate of fourier series under Holder-continuity Let$f \in C^{\alpha}(\mathbb{T})$be a holder-continuous function, i.e. $$|f(x) - f(y)| \le C|x-y|^{\alpha}$$ for some$0 < \alpha < 1$. We use$\|f\|_{C_\alpha}$to denote the smallest ... • 187 3 votes 0 answers 65 views ### Continuity of$f$and monotone decreasing of$g(x) = \cos(f(x))$implies uniform continuity of$f$Let$f,g : \mathbb{R} \rightarrow \mathbb{R}$. I want to show uniformly continuity of$f$under$f$is continuous and$f(0) = \frac{\pi}{2}$,$g(x) = \cos(f(x))$is monotonically decreasing. I know ... • 6,134 3 votes 0 answers 72 views ### If$f(x) = x^2$is uniformly continuous on open set A, then is$f$bounded? Let$A$be a nonempty set of real numbers and let$f : A \rightarrow [0,\infty)$be given by$f(x) = x ^2$. Prove or disprove that if$A$is open and$f$is uniformly continuous, then$A$is bounded. ... • 53 3 votes 1 answer 476 views ### Conditions for a function to be Lipschitz or just uniformly continuous Question. Let$\Omega$be an open subset of$\mathbb R^3$, and let$f: \Omega \to \mathbb R$be continuous. Then$f$is (a) Lipschitz, if it is$C^1$and$\nabla f$is bounded; (b) uniformly ... • 6,845 3 votes 0 answers 64 views ### Showing uniform convergence of a sequence of functions which are inherited from a different function Let$f: \Bbb R \times [0,1] \to \Bbb R$be a continuous function and$(x_n)$be a sequence in$\Bbb R$that converges to$x$in$\Bbb R$. Then show that the sequence of functions$g_ndefined by,... • 5,938 3 votes 0 answers 239 views ### Is a Function on Bounded Domain Taking Cauchy Sequences to Cauchy Sequences Uniformly Continuous? I have a question similar to this one: Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? So, from that discussion, we know that a function that takes Cauchy sequences ... • 149 3 votes 0 answers 1k views ### Prove x^n is Uniformly Continuous on a Bounded Subset of \mathbb{R} Hello I want to prove that f(x)=x^n is uniformly continuous on any bounded subset of \mathbb{R}. I'm wondering if my proof is correct, and I'm primarily wondering if my choice of \delta is ... 3 votes 0 answers 107 views ### Is it okay if I considered f as uniformly continuous and bounded on specific interval if f is continuous? Problem Suppose f is continuous and \phi is of bounded variation on [a, b]. Show that the function \psi(x)=\int_a^xfd\phi is of bounded variation on [a, b]. If g is continous on [a, b]... • 3,171 3 votes 0 answers 50 views ### Convergence in the product of spaces of iteratively composed functions. My question is a bit odd, in fact conceptually it is not difficult, only that it operates on objects that are complex (to me). I would like to check two types of convergence in the product of the ... • 327 3 votes 0 answers 683 views ### Proof of uniform continuity on compact sets Show that a function f:\mathbb{R} \rightarrow \mathbb{R} that is continuous on a compact set K is uniformly continuous on K. Is the proof below correct? Proof: Let \epsilon > 0 and let ... • 916 2 votes 0 answers 28 views ### STFT uniform continuity I want to show that short-time Fourier transform of f \in L^2 w.r.t g \in L^2 \begin{align*} \mathcal{V}_{g} f (x, \omega) = \int\limits_{\mathbb{R}}^{} f(t) \overline{g(t-x)} e^{-2 \pi i t \omega}... • 893 2 votes 0 answers 70 views ### What is the sence of "f(x,y) is continuous at x_0 uniformly in y"? Consider a function f(x,y): \mathbb{R}^2 \to \mathbb{R}. I found the following phrase in a book: "f(x,y) is continuous at x_0 uniformly in y". What does it mean? I think that it ... 2 votes 3 answers 100 views ### Uniform continuity of the function f(x)=x^a Prove that f:[0, \infty ) \to [0, \infty), f(x)=x^a with a>0 is uniformally continous if and only if 0<a\leq 1. I guess that we could start with the scratch work like this: \forall \... 2 votes 0 answers 40 views ### Proving 1/x is uniformly continuous on [1/2,1] We wish to prove that f(x)=\frac{1}{x} defined on [1/2,1] is uniformly continuous. Scratch work: Take x,y \in [1/2,1], |f(y)-f(x)|=|\frac{1}{y}-\frac{1}{x}|=|\frac{y-x}{xy}| At most x=1,y=1 ... 2 votes 0 answers 48 views ### Proof of uniform continuity of f(x) = \inf\{\mid x - a \mid : a \in S\}, S \subset \mathbb{R} I was wondering if my proof of the uniform continuity of f(x) = \inf\{| x - a | \colon a \in S\}, S \subset \mathbb{R} was correct. Attempt: | f(x) - f(y) | = | \inf\{|(x-a)|\} - \inf\{|(y-a)|\} |... 2 votes 0 answers 51 views ### Prove sin(x) is uniformly continuous on R with Heine-Cantor Theorem I've already proven uniform continuity of f(x)=sin(x) on R via epsilon-delta, but I wanted to try and prove it with the Heine-Cantor-Theorem since it seems more intuitive: Now obviously with the ... • 23 2 votes 0 answers 32 views ### Is the follwoing function an example of an uniformly continuous function? Is id an uniformly continuous function? Let (\mathbb{N},d) be a metric space such that d(x,y) = |\frac{1}{x}-\frac{1}{y}|. Let us consider the identity map id:(\mathbb{N},d) \to (\mathbb{N},d_{... • 3,392 2 votes 0 answers 51 views ### Alternative proof that a continuous function on metric spaces with compact domain is uniformly continuous I just came up with the following proof. What I like about it is that when I came up with it I felt like I was just following my nose. It's somewhat different from the other ones I've seen, so I'd ... • 1,138 2 votes 0 answers 64 views ### Non-trivial metric space X with all of C(X) uniformly continuous. Is there a metric space X such that any f\in C(X) is uniformly continuous, but there is a metric space Y and g:X\to Y which is continuous but not uniformly continuous? If X is compact or ... • 6,192 2 votes 0 answers 65 views ### What properties does this function have? Let f\colon\mathbb{R} \rightarrow \mathbb{R} be an \textbf{uniformly continuous} function. Then by definition \forall\epsilon>0 \quad \exists\delta>0\quadsuch that \forall x,y\in\mathbb{... • 103 2 votes 0 answers 81 views ### Uniformly continuous f such that \lim_{n \to +\infty}f(nx) = 0 \implies \lim_{x \to +\infty} f(x)= 0 Let f : \mathbb R \to \mathbb R a uniformly continuous function such that \forall x > 0\lim_{n \to +\infty}f(nx) = 0$$Show that$$\lim_{x \to +\infty} f(x)= 0$$I cannot use Baire Category ... • 514 2 votes 0 answers 46 views ### There exist a point c \in [a,b] such that f(c)= \frac{f(x_{1})+f(x_{2})+...+f(x_{n})}{n}. Let f: [a,b]\to \mathbb{R} be continuous on [a,b] and x_{1},x_{2},...,x_{n} \in [a,b].Then there is a point c \in [a,b] such that f(c)= \frac{f(x_{1})+f(x_{2})+...+f(x_{n})}{n}. Can ... • 1,087 2 votes 0 answers 65 views ### Prove that f:X\rightarrow\Bbb{R} is uniformly continuous iff it maps equivalent sequences onto equivalent sequences Let X be a subset of \Bbb{R}, and let f:X\to\Bbb{R} be a function. Then the following two statements are logically equivalent: (a) f is uniformly continuous on X. (b) Whenever (x_{n})_{n=... • 21.4k 2 votes 0 answers 51 views ### Does the limit of this integrand (product of monotonic functions) exist, given the integral is bounded? For p, f sufficiently differentiable functions: \begin{equation} \int_{-\infty}^{\infty} f^2 p = 1 \qquad \text{(i.e. f \in L^2_p)} \end{equation} I want to conclude \lim_{x\to\pm\infty} f^2 p ... • 391 2 votes 0 answers 28 views ### \|T_af-f\|_\infty\rightarrow 0,\ a\rightarrow 0\Rightarrow\ f has uniformly continuous representative Let f\in L^\infty(\mathbb{R}^d) (where the measure we're working with is the Lebesgue measure) and I have to prove the following equivalence : f has a uniformly continuous representative if and ... • 996 2 votes 0 answers 55 views ### Proving uniform continuity for x/(1+x) on [0,\infty) I want to show that f(x) = x/(1+x) is uniform continuous on [0, \infty). My proof: Let x,y be real numbers in [0, \infty) and let \epsilon be a positive number. We choose \delta = \... • 169 2 votes 1 answer 174 views ### Definition of uniformity in different contexts My question is rather a semantic one. I am wondering what uniformity in different contexts in general means. I know what the definition of uniform continuity is. Has uniformity in general something ... • 121 2 votes 0 answers 51 views ### Is f:\mathbb{R}^{2} \backslash E \times\{0\}\to\mathbb{R} uniform continious function? Can you help me with this problem? Let E\subseteq \mathbb{R}, U=\mathbb{R}^{2} \backslash E \times\{0\}. Prove that the following statements are equivalent: 1)Any continious function f:U\to\... • 93 2 votes 0 answers 34 views ### Extraneous Condition in the Hypothesis? If \{f_n\} is a sequence of continuously differentiable functions on [a,b], f_n \to f uniformly on [a,b[, and there is a function g : [a,b] \to \Bbb{R} such that f'_n \to g uniformly on [... • 7,370 2 votes 0 answers 148 views ### Prove that, any continuous and periodic function on \Bbb{R} is uniformly continuous on \Bbb{R}. There is a hint in my book to solve the problem. It goes as follows- Suppose, f:\Bbb{R}\to\Bbb{R} is continuous and has the period p\in\Bbb{R}. Hence, f(x+np)=f(x)\forall x\in\Bbb{R}, \: \... • 2,673 2 votes 0 answers 53 views ### Reference for theorem bounding distances of values of uniformly continuous functions? I think I proved the following theorem, but the proof is pretty long. So I thought maybe there is a reference for it because the statement seems standard. Does anyone know such a reference? Thank you! ... • 569 2 votes 0 answers 40 views ### Existence of smallest \delta>0 for a continuous function There's something on my textbook that I don't understand: Let f:I \to \mathbb{R} be continuous on the interval I.So \forall x_0 \epsilon I and \forall \epsilon>0 there \exists ... 2 votes 0 answers 50 views ### Uniformly Convergent Series 1 Fourier Coefficients Let f be an absolutely integrable function on (0,2 \pi). We have F(x) := \int_0^x f(t) \, dt,$$F(x)= \frac{\alpha_0}{2} x + \sum_{k=1}^\infty \frac{\alpha_k \sin k x + \beta_k ( 1 - \cos k x ) ... • 9,110 2 votes 0 answers 265 views ###X$a connected space ; if all continuous functions from$X$to$\Bbb R$are uniformly continuous, then X is a compact Let$X$be a subset of a normed vector space.$X$is also a connected space. Show that if all continuous functions from$X$to$\Bbb R\$ are uniformly continuous, then X is a compact. Need some help ; ... 