Intereting Posts

Representation of a number as a sum of squares.
Iff Interpretation
Sequence of random variables with stopping rule
Does every automorphism of a permutation group preserve cycle structure?
Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
Faster arithmetic with finite continued fractions
Estimates for the normal approximation of the binomial distribution
Generating connected module over a connected $K$-algebra
How do we find the inverse of a function with $2$ variables?
What is a lift?
Is composition of measurable functions measurable?
How to Classify $2$-Plane Bundles over $S^2$?
What does Structure-Preserving mean?
(Fast way to) Get a combination given its position in (reverse-)lexicographic order
Computing the Integral $\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$

Suppose $f : (a,b) \to \mathbb{R}$ such that $f’$ exists and is bounded on $(a,b)$.

Then is $f$ uniformly continuous?

I have a hint to use the mean value theorem, but I’m not sure I can apply it to an open interval like this? Does the bounded derivative imply that $f$ is continuous on $[a,b]$ somehow?

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- Relationship Between Ratio Test and Power Series Radius of Convergence
- How can I pick up analysis quickly?
- ODE $d^2y/dx^2 + y/a^2 = u(x)$

- Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another
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- Proving if $\limsup x_n = \liminf x_n = c$, then $x_n \rightarrow c, n \rightarrow \infty$ using $\epsilon$
- Continuity and Joint Continuity
- Why can't calculus be done on the rational numbers?
- A sequence of singular measures converging weakly* to a continuous measure
- If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
- What's the definition of $C^k(\overline\Lambda)$ for a bounded and open $\Lambda\subseteq\mathbb R^d$?
- Show $\max{\{a,b\}}=\frac1{2}(a+b+|a-b|)$
- Completing the differential equation from exercise 10.23 in Tom Apostol's “Mathematical Analysis”

Use the MVT to check that $f$ is a Lipschitz function.

Let $x,y \in (a,b)$. By the MVT, there is $z \in (a,b)$ between $x$ and $y$ with $f(x)-f(y) = f'(z)(x-y)$. Since $|f’| \leq M$ for some $M \geq 0$, we have: $$|f(x)-f(y)| = |f'(z)(x-y)| \leq M|x-y|.$$

Then check that every Lipschitz function is uniformly continuous.

Let $\epsilon > 0$ and choose $\delta = \epsilon/M > 0$. So, if $|x-y| < \delta$, we get: $$|f(x)-f(y)| \leq M|x-y| < M \frac{\epsilon}{M} = \epsilon,$$and $\delta$ does not depend on the points $x$ and $y$.

- Introductory text for calculus of variations
- When random walk is upper unbounded
- Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$
- Understanding the derivative as a linear transformation
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- If a subring of a ring R has identity, does R also have the identity?
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- Approximation of $e^{-x}$
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- Primal- degenerate optimal, Dual – unique optimal