Let $f:[a,b] \to \mathbb R$ be continuous , differentiable function ( having one sided derivatives at $a$ and $b$ ) such that $f'$ is bounded on $[a,b]$ ; then is $f'$ Riemann integrable on $[a,b]$ ?
2025-01-13 00:12:04.1736727124
$f:[a,b] \to \mathbb R$ be continuous , differentiable function such that $f'$ is bounded on $[a,b]$ ; then is $f' \in \mathcal R[a,b]$?
125 Views Asked by user228168 https://math.techqa.club/user/user228168/detail At
1
There are 1 best solutions below
Related Questions in REAL-ANALYSIS
- Proving whether the limit of a sequence will always converge to 0?
- Limit of $(5n^2+2n)/(n^2-3)$ using limit definition
- If $\inf f = f(a)$, then $\exists b,c$, $f(b) = f(c)$
- Trying to prove if $S$ is a subset of $R$, every adherent point to $S$ is the limit of a sequence in $S$
- ODE existence of specific solutions
- equivalent definitions of weak topology on a topological vector space
- Bounded derivative implies uniform continuity on an open interval
- Inf and Sup question
- how to prove sup(A) where A={(n+1)/n|n∈N}?
- how to use epsilion-delta limit definition to answer the following question?
Related Questions in DERIVATIVES
- Help in finding error in derivative quotient rule
- Function satisfing : $h(x)=f(2x-1)$ with $f'(-1)=0 $ and $f'(2)=-2$ then what is $h'(x) $?
- Using the chain rule of differentiation to evaluate an integral along a curve
- Derivative of power series
- What does the second value of `x` mean here?
- Partial derivative of composition with multivariable function.
- How to take the derivative of $Y=\log(x+\sqrt{a^2+x^2})$?
- The derivative of a two-to-one complex function has no zeros.
- Derivative of power series with nonnegative coefficients
- Error in logarithmic differentiation of $R(s)=s^{\ln s}$
Related Questions in CONTINUITY
- How discontinuous can the limit function be?
- Weierstrass continuity vs sequential continuity
- Functions that change definition with the type of input
- Find the number a that makes $f(x)$ continuous everywhere?
- Let $f(x) = 1$ for rational numbers $x$, and $f(x)=0$ for irrational numbers. Show $f$ is discontinuous at every $x$ in $\mathbb{R}$
- Show that $T$ is not a homeomorphism
- Is Lipschitz "type" function Continuous?
- Continuity of the function $\mathbb{R}^k \to\mathbb{R}: x\mapsto \ln(1+ \lVert x \rVert)$
- Between uniform and pointwise convergence
- Continuity of a two variables function
Related Questions in RIEMANN-INTEGRATION
- Example of distinctions between multiple integral and iterated integrals.
- "Mean Value Theorem" for a integrable and bounded function
- How much do we really care about Riemann integration compared to Lebesgue integration?
- Improper Riemann integral versus Lebesgue
- $f:[a,b] \to \mathbb R$ be continuous , differentiable function such that $f'$ is bounded on $[a,b]$ ; then is $f' \in \mathcal R[a,b]$?
- Munkres' definition of the extended integral
- Can all continuous functions on a compact set be approximated by 1-lipschitz functions?
- Prove that $\lim_{x\rightarrow \infty} \frac{1}{x}\int_0^x f(t)dt=c \mbox{.}$
- Riemann integration of $\int_{[0,a]} f + \int_{[0,f(a)]} f^{-1}$
- $F(x) = \int_x^{x+1} \sqrt {\arctan {t}}dt$ is bounded for $x \ge 0$.
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- 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?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- 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
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
Not true. The first example given appeared in
[Added note:]
Maybe some of you are at a library with inadequate archives ... or don't like reading Italian. So I will try a little harder to help.
All you need to do is take a fat Cantor set and construct your function around that. My friend James Foran included a construction of such a function on page 267 in his book
I thought this was pretty well known. It was precisely the motivation that Lebesgue gave for developing a new theory of integration. If you look in the early literature of integration theory you find a great deal of attention on "improper" integrals--how best to handle unbounded functions. Volterra's example shows that the Riemann integral doesn't handle even the bounded functions correctly.