Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
1
vote
2 answers

Computing a gradient knowing only the Directional Derivative and unit vectors.

Suppose that $f: \mathbb{R}^2 \to \mathbb{R}$ is differentiable at $p$. Also suppose that $D_uf(p)=1$ and $D_vf(p)=1$ where $u=\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $v=\left( \frac{1}{2}, \frac{-\sqrt{3}}{2} \right)$. Compute…
emka
  • 6,494
1
vote
1 answer

A function $f: X \to Y$ is continuous if and only if $ ^{−1} (C) $ is closed in $X $ for every closed set $C$ in $Y$.

Since a mapping $f$ of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $ ^{−1} (V)$ is open in $X$ for every open set $V$ in $Y$ and since a set is closed if and only if its compliment is open, $^{−1} (E^c)= [^{−1}…
Mike Langis
  • 13
1
vote
1 answer

Power Series and Sequences and Series of Functions

Observe that $e^{-x^2} = \sum_{n = 0}^{\infty} \frac{(-1)^n}{n!} x^{2n}$ for $x$ an element of the reals. Express $F(x) = \int_{0}^x e^{-t^2}\,dt$ as a power series For part 1, I don't understand the purpose, are we just suppose to look at the…
mary
  • 2,374
1
vote
1 answer

how to show $ \iint\limits_D(x\frac{\partial f }{\partial x}+ y \frac{\partial f}{\partial y}) \, dx \, dy= \frac{\pi}{2e} $?

Assume $f\in C^2(D)$, where $D=\{(x,y)\in \mathbb R^2: x^2+ y^2 \le 1\}$, if $$ \frac{\partial^2 f }{\partial x^2} + \frac{\partial^2 f }{\partial y^2} =e^{-x^2 -y^2} $$ how do I show $$ \iint\limits_D(x\frac{\partial f }{\partial x}+ y…
Farmer
  • 1,535
1
vote
1 answer

Lipschitz continuity and slope

Lipschitz continuity is defined as follows: A function is Lipschitz continuous if there exists a $K \in \mathbb R$ such that $|f(x) - f(y)| \leq K|x-y| \forall x,y \in D$ Now I was wondering if it is possible to say that if one function's Lipschitz…
TestGuest
  • 1,053
1
vote
2 answers

Proving convergence or divergence using the comparison test.

The question is to determine if the series $$\sum_{n=1}^\infty \frac{1+\cos(nx)}{n^{4}}$$ converges or diverges using the comparison test. I established that for $x=\frac{\pi}{2}, \frac{3\pi}{2},$ etc (first time using MSE so I don’t know the code…
user728655
  • 61
  • 6
1
vote
1 answer

Equations where one side may not exist

Every now and then, I encounter an equation (in analysis, usually) that needs to be interpreted as, "if one side exists, then the other side exists, and then the two sides are equal." This prompted me to ask, are there examples of equations where…
Timothy Chow
  • 752
1
vote
2 answers

each continuous function $f:X\to \mathbb{R}^2$ is bounded

Prove or Disprove : Let $X \subset \mathbb{Q}^2$. Suppose that each continuous function $f:X\to \mathbb{R}^2$ is bounded. Then $X$ is necessarily finite. I think this statement is wrong as if we know that every continuous function takes compact…
RAM_3R
  • 1,256
1
vote
2 answers

Accumulation points of $A\subseteq C([0,1])$

Consider the subset A of $C([0,1]) $ consisting of continuous functions f with $f(0)=f(1)=0$ In $(C([0,1]), ||\cdot||_1)$ determine whether the follow are accumulation points of the set A 1) $g_1(t)=0$ 2) $g_2(t)=t$ The definition I have for…
Mathsstudent147
  • 1,495
1
vote
1 answer

A question with Jordan measurable

$\textbf{Definition :}$ A set $X\in \mathbb{R}^n$ is Jordan measurable if exists $R$ a rectangle such that $A\subseteq R$ and the function $\chi_{A} : R \rightarrow \mathbb{R}$ is integrable. Consider $Y\subseteq X$ where $X,Y$ are Jordan…
user411479
1
vote
2 answers

Prove or disprove convergence $\sum (a_n)^m$

Problem : Given sequence $\left\{a_n\right\}$ : $$ \forall n\in\mathbb{N}, a_n > 0, \quad \lim_{n\to\infty}a_n=0$$ Does there always exist some positive real number $m$ which makes series $$\sum_{n=1}^\infty (a_n)^m < \infty$$ converge? I think…
bFur4list
  • 2,604
1
vote
0 answers

Suppose $f(x)$ is continuous on $[0,\,1]$, differentiable in $(0,\,1)$, and $f(0)=f(1)=0,\qquad f(\frac{1}{2})=1$.

Suppose $f(x)$ is continuous on $[0,\,1]$, differentiable in $(0,\,1)$, and $$ f(0)=f(1)=0,\qquad f(\frac{1}{2})=1. $$ Show that $\exists\;\xi\in(0,\,1)$, such that $$ f'(\xi)-3\big(f(\xi)-\xi\big)=1. $$ My idea is to get a primitive of…
Knt
  • 1,649
1
vote
0 answers

Suppose $f(x)$ is two-times differentiable, $f(x),\;f'(x),\;f''(x)$ are all $>0$ and there are $a,\,b>0$ such that $f''(x)\leq af(x)+bf'(x)$

Suppose $f(x)$ is two-times differentiable in $\mathbb R$, $f(x),\;f'(x),\;f''(x)$ are all $>0$ and there are $a,\,b>0$ such that $$ f''(x)\leq af(x)+bf'(x),\qquad\text{for all $x\in\mathbb R$}. $$ Show that (1)…
Knt
  • 1,649
1
vote
1 answer

Does the difference set of the closures equal the closure of the difference set?

Let $X$ and $Y$ be normed vector spaces and $T$ be a linear operator from $X$ to $Y$ whose norm is $1$. If $U$ is an open set in $X$, is it true that $\text{closure}(T(U))-\text{closure}(T(U))=\text{closure}(T(U)-T(U))$? Here, $A-B$ is a set of…
tuko
  • 51
  • 3
1
vote
0 answers

Existence of non-constant function $f$

let $f\colon\mathbb{R}\to\mathbb{R}$ is continuous function. For arbitrary two real number $a, b$ $(a
bFur4list
  • 2,604
1 2 3
…
99 100