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
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Metric topology: boundaries proof

I am stuck on what seems like a completely intuitive proof. A is a subset of X. ( + for disjoint union) I need to show, first, that (i) Closure of S = interior of s + boundary of S Then I am asked to show that (ii) X = interior of S + boundary of…
Juniper Leaf
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Prove central symmetry in (0.5;0.5) if f(1-x)=1-f(x)

I would like to prove that if $f:\left[0,1\right]\to\left[0,1\right]$ such that and $f\left(1-x\right)=1-f\left(x\right)$, then $f$ has a central symmetry at $\left(0.5,0.5\right)$. This is actually an intuition but I can't prove it. I did the…
Gim
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Check if the sequence converges?

Given the sequence $$\frac{1+2+...+n}{n+2}-\frac{n}{2}$$ I am asked to check if it converges. How can I do this? One way is to check if the sequence is bounded and monotonic, right? But how could we see if the sequence is monotonic?
Mary Star
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show that that the sequence $(a_{n})$ has a decreasing subsequence that converges to 0

Let $(a_{n})$ a sequence of positive numbers.It is given that $\inf A=0$,where $A=\{a_{n}:n \in N\}$.How can I show that that the sequence $(a_{n})$ has a decreasing subsequence that converges to $0$?
evinda
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which solution is the right one?

Let $0
evinda
  • 7,823
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Continuously differentiable sequence of functions

I wish to find the best way to prove that $\phi_n:= [\psi(x+{1\over n})-\psi(x)]n$ where $\psi$ is continuously differentiable on $(a,b)$, converges uniformly to $\psi'(x)$ on all closed subintervals of $(a,b)$. It is clear that $(\phi_n)$ converges…
Jesse
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Cantor's completeness principle

I hope everyone who has underwent a fundamental course in Analysis must be knowing about Cantor's completeness principle. It says that in a nest of closed intervals ,the intersection of all the intervals is a single point. I hope I can get an…
Primeczar
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Problem in walter rudin RAC

The function $f(x)=x^2\sin(\frac{1}{x})$ if $x\neq 0$ , $f(0)=0$.Then $f$ is differentiable at every point, but $\int_0^1|f^{'}(x)|dx=\infty.$ I proved $f$ is differentiable at every point. To prove $f$ is not integrable i integrate the derivative…
Pitchai murugan
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Suprema proof: Prove $\sup(f+g)\leq \sup f+\sup g$

I need help with these: 1) Show the following inequality for the supremum of functions $f:\mathbb{R}\to \mathbb{R}$ and $g: \mathbb{R}\to\mathbb{R}$ $$ \sup(f+g)(x)\leq \sup f(x)+\sup g(x) $$ 2) What can you say about a set $M$ of real numbers if…
Fred
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Prove the following subspace is closed in $L^2(\mathbb{R})$

Let $V_j=\{f\in L^2(\mathbb{R}): f$ is constant on $[\frac{n}{2^j},\frac{n+1}{2^{j}}) $for all $ n \in\mathbb{Z}\}$ , $j\in\mathbb{Z}$ be a sub set of $L^2(\mathbb{R}).$ Prove that $V_j$ is an closed linear subspace of $L^2(\mathbb{R})$. Attempt: I…
Murugan
  • 54
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Derivative of the norm of a second order tensor w.r.t. the tensor

I came across this derivative in a paper: $\frac{\partial{\|{A}\|}}{\partial A} = \frac{A}{\|A\|}$ where A is a rank 2 tensor. And I wonder what is the definition of this norm. I believe it is not a general expression. One more thing, what would be…
Vahid
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to show a function is bounded when domain is bounded

How to show that $f$ : $S$ -> $R$ is uniformly continuous and $S \subset R $ is bounded then $f$ must be bounded. I have tried using the theorem which is states the three statements to be equivalent.But could not suceed.
Rusty
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Family of Closed/Open Nested Intervals

Find a family $\{I_n\}$ of closed nested intervals, such that no two $I_n$'s are equal and their intersection is $[-2,2]$. An answer for the same question except for dealing with open nested intervals would also be appreciated.
user123652
  • 235
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Uniform continuity of $\frac{\sin (y^3)}{5+y}$ for $y \geq 0$

Is $f\colon y\mapsto\dfrac{\sin(y^3)}{5+y}$ for $y\geq 0$ uniformly continuous? I think it is but I can't seem to prove that. (Brain-dead day.) Help would be very much appreciated.
Dan
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Convex function in its interior

Let $f$ be a convex function on an open subset of $R^{n}$. How to prove $f$ is continuous in the interior of its domain. For $n=1$, let $f$ be convex on the set $(a,b)$ with $a
user107723
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