A sequence $\{ f_n\}$ of Lipschitz continuous functions (with corresponding Lipschitz constants $M_n$) has pointwise limit $f$ which is also Lipschitz continuous (with Lipschitz constant $M$). Is it true that the sequence $\{ M_n\}$ is bounded? And if not, under what extra assumptions is it true?
2026-02-25 08:55:01.1772009701
pointwise limit of sequence of Lipschitz continuous functions is Lipschitz continuous $\Rightarrow$ Lipschitz constants bounded?
371 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in SEQUENCES-AND-SERIES
- How to show that $k < m_1+2$?
- Justify an approximation of $\sum_{n=1}^\infty G_n/\binom{\frac{n}{2}+\frac{1}{2}}{\frac{n}{2}}$, where $G_n$ denotes the Gregory coefficients
- Negative Countdown
- Calculating the radius of convergence for $\sum _{n=1}^{\infty}\frac{\left(\sqrt{ n^2+n}-\sqrt{n^2+1}\right)^n}{n^2}z^n$
- Show that the sequence is bounded below 3
- A particular exercise on convergence of recursive sequence
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Powers of a simple matrix and Catalan numbers
- Convergence of a rational sequence to a irrational limit
- studying the convergence of a series:
Related Questions in FUNCTIONS
- Functions - confusion regarding properties, as per example in wiki
- Composition of functions - properties
- Finding Range from Domain
- Why is surjectivity defined using $\exists$ rather than $\exists !$
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Lower bound of bounded functions.
- Does there exist any relationship between non-constant $N$-Exhaustible function and differentiability?
- Given a function, prove that it's injective
- Surjective function proof
- How to find image of a function
Related Questions in LIPSCHITZ-FUNCTIONS
- Is a Lipschitz function differentiable?
- Equivalence for a reversed Lipschitz-type condition
- Compact sets in uniform norm
- Does locally Lipschitz imply Lipschitz on closed balls?
- An upper bound for $\|2 \nabla f(x) - \nabla f(y)\|$ in terms of $\|x-y\|$ if the gradient is $L$-Lipschitz
- Nowhere-differentiable Lipschitz-continuous function
- How to prove the following function is not Lipschitz continuous?
- Question on Lipschitz continuity
- Is the Borel isomorphic interchanging-digit map a k-Lipschitz map?
- Could lower semicontinuous functions have Lipschitz constant?
Related Questions in POINTWISE-CONVERGENCE
- I can't understand why this sequence of functions does not have more than one pointwise limit?
- Typewriter sequence does not converge pointwise.
- Fourier Series on $L^1\left(\left[0,1\right)\right)\cap C\left(\left[0,1\right)\right)$
- Analyze the Pointwise and Uniform Convergence of: $f_n(x) = \frac{\sin{nx}}{n^3}, x \in \mathbb{R}$
- Uniform Convergence of the Sequence of the function: $f_n(x) = \frac{1}{1+nx^2}, x\in \mathbb{R}$
- Elementary question on pointwise convergence and norm continuity
- Pointwise and Uniform Convergence. Showing unique limit.
- A sequence $f_k:\Omega\rightarrow \mathbb R$ such that $\int f_k=0 \quad \forall k\in \mathbb N $ and $\lim\limits_{k\to\infty} f_k \equiv1$.
- Show that partial sums of a function converge pointwise but not uniformly
- example of a sequence of uniformly continuous functions on a compact domain converging, not uniformly, to a uniformly continuous function
Related Questions in SEQUENCE-OF-FUNCTION
- Convergence in measure preserves measurability
- Analysis Counterexamples
- Arzelá-Ascoli Theorem precompact sets
- Uniform limit not being equal to pointwise limit?
- $C^\infty_0$ approximation of $L^\infty$
- Understanding Uniformly Cauchy
- Proving that this function converges uniformly.
- Thinking of sequence where $f_n'$ does not converge to $f'$
- Rudin proof change, 7.27.
- The sequence $\{n(n-1)\}$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
No. Let $f_n(t)=\sin(n^2t)/n$.