Intereting Posts

Numerical Analysis over Finite Fields
Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
Materials for self-study (problems and answers)
Is the number 0.2343434343434.. rational?
How gaussian mixture models work?
Easy way to show that $\mathbb{Z}{2}]$ is the ring of integers of $\mathbb{Q}{2}]$
Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.
Tables of Hypergeometric Functions
Finding a unit vector perpendicular to another vector
Relative homology groups
Problem 7 IMC 2015 – Integral and Limit
Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin…(\sin(x)..)$
Most efficient way to integrate $\int_0^\pi \sqrt{4\sin^2 x – 4\sin x + 1}\,dx$?
Proof of the inequality $(x+y)^n\leq 2^{n-1}(x^n+y^n)$
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.

Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be uniformly continuous with $g(0)=0,c\geq 0, c \in \mathbb{R}$. Show: $$\exists a\geq 0 \in \mathbb{R}: \forall x \in \mathbb{R}: |g(x)| \leq a \cdot |x|+c$$

I could also say $g(x) \in \mathcal{O}(x)$.

Notes: I could not make up any counterexample so I guess it could be true, all uniformly continuous functions I know grow too slowly.

- Suppose f is differentiable on an interval I. Prove that f' is bounded on I if and only if exists a constant M such that $|f(x) - f(y)| \le M|x - y|$
- Writing the roots of a polynomial with varying coefficients as continuous functions?
- Uniform convergence, but no absolute uniform convergence
- Given $\lim\limits_{x\to a}{f^\prime(x)}=\infty$, what can be concluded about $f(a)$?
- Continuity of a series of functions
- Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}$

My approach:

Given $\epsilon > 0$, we have that: $$\exists \delta(\epsilon): |x-y|<\delta=> |g(x)-g(y)|<\epsilon$$

because of the continuity of $g$. Now choose $n=\text{max}\{n \in \mathbb{N}: (n-1)\delta/2\leq|x|\}$. Obviously, such an $n$ exists, and $n > 0$. We also easily see that an upper bound for $n$ is $n \leq \frac{2}{\delta}|x|+1$.

Now we use this to separate $|x|$ into $n-1$ distinct parts of size $s<\delta/2$, and the last part which is smaller than $\delta$ :

$$|x|=|x_1-x_0|+|x_2-x_1|+|x_3-x_2|+…+|x_n-x_{n-1}| < (n-1)\delta/2 + \delta = (n+1)\delta/2.$$

$$\begin{align}

\Rightarrow |g(x)| & =|g(x_1)-g(x_0)+g(x_2)-g(x_1)+g(x_3)-g(x_2)+…+g(x_n)-g(x_{n-1})| \\

& \leq |g(x_1)-g(x_0)|+|g(x_2)-g(x_1)|+|g(x_3)-g(x_2)|+…+|g(x_n)-g(x_{n-1})| \\

& \lt n \cdot \epsilon \leq (\frac{2}{\delta}|x|+1) \cdot \epsilon = \frac{2\epsilon}{\delta} \cdot |x|+\epsilon

\end{align}$$

So we can see that the constant $c$ we were given can be set as the $\epsilon := c$, and that was also the reason why generally speaking $c>0$. Then we can choose $a := \frac{2\epsilon}{\delta}$, as our $\delta$ only depends on the $\epsilon$, and we have that $|g(x)| \leq a \cdot |x| + c$ for $c > 0$. $\quad \square$

- How to evaluate $\sum\limits_{k=0}^{n} \sqrt{\binom{n}{k}} $
- Continuity of a series of functions
- Is a bounded and continuous function uniformly continuous?
- Convergence of $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$?
- Does it make any sense to prove $0.999\ldots=1$?
- Does a closed and bounded set in $\mathbb{R}$ necessarily contain its supremum and infimum?
- Is there a proof that there is no general method to solve transcendental equations?
- How to prove that the rank of a matrix is a lower semi-continuous function?
- A necessary condition for series convergence with positive monotonically decreasing terms
- How does one prove the Taylor's Theorem by the Cauchy's Mean Value Theorem?

It is false if $c=0$. To see this, try to think of a continuous function that grows very rapidly near $0$.

It is true if $c\gt 0$. One way to show it is by taking a number of very small steps from $0$ to $x$, small enough to guarantee (using uniform continuity) that the function changes no more than a certain fixed amount at each step. Trying to write out the details should lead you to what this fixed amount is, and to what value of $a$ will work.

- Why is compactness in logic called compactness?
- Closed form solutions for a family of hypergeometric sums.
- Derivative of a function is the equation of the tangent line?
- Is $7$ the only prime followed by a cube?
- Alternate ways to prove that $4$ divides $5^n-1$
- Show that $SL(n, \mathbb{R})$ is a $(n^2 -1)$ smooth submanifold of $M(n,\mathbb{R})$
- A basis for the dual space of $V$
- Integral dx term sanity check
- Simple theorems that are instances of deep mathematics
- every field of characteristic 0 has a discrete valuation ring?
- Proving the equivalency of Principle of Mathematical Induction and Well Ordering Principle
- How many idempotent elements does the ring ${\bf Z}_n$ contain?
- How to prove that $\,\,f\equiv 0,$ without using Weierstrass theorem?
- Proving compactness of $\{0\}\cup\{\frac1n; n\in\mathbb N\}$ by definition
- holomorphic function on punctured disk satisfying $\left|f\left(\frac{1}{n}\right)\right|\leq\frac{1}{n!}$ has an essential singularity at $0$