Intereting Posts

Matrices that commute with all matrices
Find $\lim_{x \to – \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$
Quadratic extensions of $\mathbb Q$
Why are “algebras” called algebras?
How do we prove this asymptotic relation $\lim\limits_{k\rightarrow \infty}G(k)=\sqrt{2}$
Can this inequality proof be demystified?
Ambiguous matrix representation of imaginary unit?
Is there an approximation to the natural log function at large values?
Check whether a point is within a 3D Triangle
Question about trigonometry/trigonometry question?
Check Points are line, triangle, circle or rectangle
Equivalences to “D-finite = finite”
Finitely generated graded modules over $K$
Show that $\lim_{n\to \infty} (\sqrt{n+1} – \sqrt n) = 0$ using the definition of a limit
Need help understanding Erdős' proof about divergence of $\sum\frac1p$

Can anyone help?

Let $g:I \to \mathbb{R}$ be an uniformly continuous function, where $I$ is an interval. Prove that exists an constant $c$ that satisfies:

$$\lvert g(x)-g(y)\rvert < 1 + c \lvert x-y \rvert, \forall x,y \in I$$

- How to prove $r^2=2$ ? (Dedekind's cut)
- Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$
- Cardinality of set of real continuous functions
- A converse proposition to the Mean Value Theorem
- Is the indicator function of the rationals Riemann integrable?
- Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.
- What do physicists mean with this bra-ket notation?
- An inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$
- Inverse function theorem application
- On the existence of a particular solution for an ODE

Apply the suggestion of @5pm. Take $\delta$ in the definition of uniformly continuity which corresponds to $\epsilon=1$. You can suppose $\delta<1$ (decreasing $\delta$ does not hurt). Given $x,y\in I$ let $x=x_0<x_1<\dots <x_n = y$ so that $|x_{i+1} – x_i|< \delta$. You can do this with $n = \lceil (x-y)/\delta \rceil$, so

$$

|f(x)-f(y)| \le \sum_{i=1}^n |f(x_{i})-f(x_{i-1})| \le n = \lceil (x-y)/\delta \rceil \le 1 + (x-y)/\delta.

$$

Let $c=1/\delta$ and you are done.

From the definition of uniform continuity you have that $|x-y|<\delta$ implies $|g(x) – g(y)| < \epsilon$ and specifically $\delta$ can only depend on $\epsilon$ and must be independent of $x$ and $y$. So you would have that $|x-y|< M\epsilon$ where $M \in \mathbb{R}$ for all $x,y \in I$ implies $|g(x) – g(y)| < \epsilon$.

From here I’m not entirely sure where to go, I’m wondering if you can assume $\epsilon <1$ so you get $1 + \frac{1}{M}|x-y| < 1 + \epsilon$ but since, from the definition of continuity $\epsilon > 0$ you get $1 <1 + \frac{1}{M}|x-y| < 1 + \epsilon$

Thus

$|g(x) – g(y)| < 1 + \frac{1}{M}|x-y|$

Where $\frac{1}{M}$ is your constant.

There’s far more qualified people on here though (I’m just nervous about posting it, in case it takes you in a wrong direction)

- Maximum of $\frac{\phi(i)}i$
- Limit of Lebesgue integral in $L_1(,m)$
- Sum of $k {n \choose k}$ is $n2^{n-1}$
- Proving that the terms of the sequence $(nx-\lfloor nx \rfloor)$ is dense in $$.
- find maximum area
- Two CW complexes with isomorphic homotopy groups and homology, yet not homotopy equivalent
- Examples proving why the tensor product does not distribute over direct products.
- Numbering edges of a cube from 1 to 12 such that sum of edges on any face is equal
- Invariant subspaces if $f$ is defined by more than one matrix
- Vector equation and parametric equation for a line segment
- The fundamental group of some space
- $A,B\in M_{n}(\mathbb{R})$ so that $A>0, B>0$, prove that $\det (A+B)>\max (\det(A), \det(B))$
- Why did my friend lose all his money?
- Multivariate Taylor Series Derivation (2D)
- Why do we look at morphisms?