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
2
votes
2 answers

Prove: If the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then f must be a constant function

Prove: If the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then $f$ must be a constant function on $[a,b]$. I need to select some $x_1$ and $x_2$ in $[a,b]$ such that $x_1$ is not equal to $x_2$…
Mary
  • 149
2
votes
0 answers

Weierstrass theorem.

Prove that for the function $f\in C([0,1])$ $\forall x\in[0,1]:$ $\sum_{k=0}^n \binom{n}{k}x^k(1-x)^{n-k}(-1)^kf(\frac {k}{n}) \to0$, $n\to \infty$. I think we can use a Bernstein polynomial, but how?
Ihor
  • 31
2
votes
2 answers

Show $x^2$ in the interval $(0,1/3]$ has no fixed points.

Show $x^2$ in the interval $(0,1/3]$ has no fixed points. I understand that the range of that domain is always lower than $y=x$, but what is a proper way of showing this? $$\left(0,\frac13\right] \to \left(0,\frac19\right]$$
Mary
  • 149
2
votes
3 answers

continuous and strictly increasing implies differentiable

I am not sure if this is true, but intuitively it seems that if a function is strictly increasing and it is also continuous...it is differentiable. It may be because there are no bumps like in the absolute value.
Usman
  • 39
2
votes
2 answers

Pointwise Convergence Implies Uniform?

Let $f_{n} \to 0$ pointwise on the interval $[-A,A]$ where $f_{n}$ are continuous and uniformly bounded. Can we show the convergence is uniform? Thanks, any help appreciated.
Red Rover
  • 847
2
votes
1 answer

Completion of a metric space

Let $R$ be a topological ring (i.e addition and product are continuous) which we assume it is metrizable with metric d and consider the completion $\hat{R}$ of the ring $R$ defined as the set of all classes of equivalent Cauchy sequences. Question:…
Mertier R.
  • 23
2
votes
1 answer

Continuous at some and at all points

Can I have someone to show me an insight on how to prove these? I had referred to a number of books but most authors merely state them as definitions or theorems without proof. Let $T:X \to Y$ be a linear operator. Then the following are…
Sandra
  • 563
2
votes
1 answer

A simple question about definition of Legendre's transform

Let a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be convex and satisfies $$ \lim_{|q| \rightarrow \infty}\frac{f(q)}{|q|}=+\infty.$$ The Legendre's transsformation of $L$ is defined by $$L^*(p)=sup_{q \in\mathbb{R}^n} [p\cdot q-L(q)]$$ for $p…
Richard
  • 4,432
2
votes
1 answer

Discontinuous surjective functions with continuous and discontinuous compositions

Do there exist two discontinuous surjective functions $f,g: \mathbb{R} \to \mathbb{R}$ such that the only one composition of them $f(g(x))$ or $g(f(x))$ is continuous? I tried to use $x$ for rational and $-x$ for irrational numbers, and something…
Matt24519
  • 21
2
votes
1 answer

Prove that the function is continuous at x0 = 0.

Hi guys I have this question here from my assignment and here's my attempt at the above question (apologies for the handwriting) I wonder is it possible if someone could tell me if I have the right approach or maybe to try something different.…
phil demeanour
  • 69
2
votes
3 answers

Prove that the following is a constant function

Let $f : R \rightarrow R $ $\lvert f(x)-f(y) \rvert \le (x-y)^2, \forall x,y \in R $ Any sort of help is appreciated! I know I am not suppose to ask for the entire solution, so I will ask for strong hints.
Trying2LearnMath
  • 123
2
votes
3 answers

Continuous and uniformly continuous proof

How would you show that if a continuous function $f:[0,1) \to \mathbb{R}$ satisfies $f(x) \to 0$ as $x \to 1$ then it is uniformly continuous?
user26069
2
votes
1 answer

Example of continuous curve $f:[0,1]\to\mathbb{C}$ for which $f(0)=0,f(1)=1$ which has no point which satisfy certain conditions?

Does there exist any continuous curve $f:[0,1]\to\mathbb{C}$ for which $f(0)=0,f(1)=1$ and for which there is no pair of points $p,q\in f([0,1])$ such that $q-p=0.75$?
user233198
  • 177
2
votes
1 answer

Corollary of Inverse Function Theorem?

Recall the inverse function theorem: Theorem. Lef $U\subseteq \mathbb R^n$ be an open subset, $p\in U$ and $f\in C^1(U, \mathbb R^n)$. If $Df(p)\in \textrm{Aut}(\mathbb R^n)$ there exists an open subset $V\subseteq U$ containing $p$ such that…
PtF
  • 9,655
2
votes
2 answers

Is $C_0(\mathbb R)$ separable?

Let $C_0$ be the Banach space of all continuous real value functions whose limits in $\pm \infty$ is zero, with the supremum norm. Is this space separable?
A.B
  • 1,536
1 2 3
…
99 100