Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with real-analysis
Search options not deleted
user 6266
For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.
1
vote
Accepted
Why is $b^x \overset{\mathrm{def}}{=} \sup \left\{ b^t \mid t \in \Bbb Q,\ t \le x \right\}$...
Probably it is easier to accept the definition of $b^x$ as the unique element of separation of the two sets
$$
B(x)=\{b^t|t\in\mathbb{Q},t\leq x\}\\
C(x)=\{b^t|t\in\mathbb{Q},t\geq x\}
$$
which implie …
- 11.1k
1
vote
Describing the rotation speed of an automatic pizza saucer
If the saucer arm produces a stripe of sauce whose width is $d$, and move toward the center with constant speed $v$, then when the disc make a $2\pi$ round, the arm should move of a distance $d$, so
$ …
- 11.1k
2
votes
1
answer
166
views
What is the English name of these definitions?
An italian real analysis book frequently uses the following definitions
Definition 1. Separated sets, separation elements.
The sets $A,B\subset\mathbb{R}$ are said to be separated if
$$a\leq b\quad\f …
- 11.1k
1
vote
Accepted
How to show injectivity of a multivariable function
This is obviously injective, because the system of equations
$$
\left\{
\begin{align}
\xi &= \frac{x}{1+x+y},\\
\eta &= \frac{y}{1+x+y},
\end{align}
\right.
$$
has a unique solution given by
$$
\left …
- 11.1k
1
vote
Accepted
Finding saddle point from the critical points
The first order derivatives, evaluated in the two know critical points are
\begin{alignat}{2}
&\frac{\partial f}{\partial x}(4,0) &&= 48-8(\alpha+\beta),\\
&\frac{\partial f}{\partial y}(4,0) && …
- 11.1k
2
votes
Accepted
Solutions for $\cos(\alpha)+\cos(\beta)-2\cos(\alpha+\beta)=0$ with a certain value range.
Rewrite the equation as
$$
\cos (\alpha )+\cos (\beta )-2 \cos (\alpha ) \cos (\beta )+2 \sin (\alpha ) \sin (\beta )=0
$$
use the Weierstrass substitution for $\beta$ (or for $\alpha$), with $t=\tan( …
- 11.1k
1
vote
Approximate piecewise constant function with continuous function
I think is difficult, or perhaps impossible, to avoid extrema of the built function.
Take
$$
y_0=a_0,\quad y_i=\frac{a_{i-1}+a_i}{2},\quad y_n=a_{n-1}
$$
then take a quadratic that pass through the re …
- 11.1k
0
votes
Accepted
Homogeneous Function Exchange of Variable
It seems correct to me. For example if
$$
f(x)=kx
$$
then
$$
\frac{d}{dx}yf(x)=\frac{d}{dx}ykx=ky=f(y)
$$
Euler theorem in this case can be expressed as
$$
x\frac{d}{dx}f(x)=f(x)
$$
and it can be appl …
- 11.1k
1
vote
Analysis - Uniform Continuity
Try to think to a continuous and differentiable function on a compact interval $[a,b]$, except for an infinite derivative on one of $a$ or $b$.
Then consider the same function on $(a,b)$.
- 11.1k
1
vote
Show that if $a_n+b_n+c_n\to0$ then $a_n,b_n,c_n$ converge to $0$
To prove that $x_n$ converges to $0$ you need to take the limit
$$
0\leq x_{n+1}\leq \frac{2}{3}x_n \implies 0\leq l\leq \frac{2}{3}l \implies l=0
$$
- 11.1k
4
votes
1
answer
664
views
Cesàro mean for a divergent sequence
Given a real sequence $(a_n)_n$ converging to a finite value $a$, a property of the Cesàro mean, defined as the arithmetic mean
$$
b_n=\frac{a_1+\ldots+a_n}{n},
$$
is
$$
\lim_{n\to\infty}b_n=a,\tag …
- 11.1k
4
votes
Accepted
Question on Functions of Bounded Variation
$$
V_f[a,b]\geq\left|f(x)-f(a)\right|+\left|f(b)-f(x)\right|\geq\left|f(x)-f(a)\right|\geq\left|\left|f(x)\right|-\left|f(a)\right|\right|
$$
- 11.1k
1
vote
Accepted
Differentiability of $\cos^{-1}(1-x^2)$ and $\cos^{-1}(1-x^4)$
I think the reasoning is correct, and the limits to calculate the right and left derivatives of the first function in $x=0$ are not so different from the one used in the second function:
\begin{align} …
- 11.1k
2
votes
Accepted
Circular definition of tangent line and derivative
Given that
$$
|L(x)-K(x)|=|L(x)-f(x)+f(x)-K(x)|\leq|f(x)-L(x)|+|f(x)-K(x)|\tag1,
$$
if
$$
|L(x)-K(x)|\geq2|f(x)-L(x)|\tag2
$$
then
$$
2|f(x)-L(x)|\leq|f(x)-L(x)|+|f(x)-K(x)|,
$$
from which you obtain …
- 11.1k
1
vote
Is it possible to differentiate $\sin x$ with respect to $\cos x$ from first principles?
Set $y=\cos x$, then, for $x\in[0,\pi]$,
$$
\frac{d\sin x}{d\cos x}=\left.\frac{d\sin(\arccos y)}{dy}\right|_{y=\cos x}=-\left.\frac{y}{\sqrt{1-y^2}}\right|_{y=\cos x}=-\frac{\cos x}{\sin x}=-\cot x,
…
- 11.1k