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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If we don't accept axiom of choice, can we disprove Heine's theorem?

Heine's theorem stated that $\lim_{x\to a}f(x)=b$ if and only if: for any sequence ${a_n},\lim_{n\to \infty}a_n=a,a_n \neq a$ ,we have$$\lim_{n \to \infty}f(a_n)=b$$ I have just learned that this theorem can be used when we accepted axiom of…
J.Guo
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Perturbing a path

Suppose $\gamma : [0,1] \to \mathbb{R}^n$ is a smooth path, such that $\gamma([0,1]) \subseteq U$, where $U$ is an open subset of $\mathbb{R}^n$. Now I would like to consider the path $\gamma_\varepsilon : [0,1] \to \mathbb{R}^n$, where…
TheGeekGreek
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Using Theta Notation $\Theta$ how can I prove that $(13n + 3)(9n + 1)(\log(4n^2 + 100))$ is an element of $\Theta(n^2 \log n)$

I'm having a hard time figuring out what systematic approach I need to follow to solve questions like these :/ Do I expand and use the Max() principle to show that it reduces to the RHS? Please help :(
Katie Rose
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Explanation of composition of two onto functions?

My book says that if functions $f$ and $g$ are both onto then $f\circ g$ and $g\circ f$ may or may not be onto. Why is this so? Would someone please help me understand this, maybe with an example or diagrammatically? My book states that$ f\circ g$…
Hema
  • 1,329
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Distance to the unit ball

I am currently reading a book on analysis. I have some finite dimensional Hilbert space $H$ and let $B$ be its unit ball. Then it says that the distance of some random point $x \in H$ to the unit ball is given by $$ d(x,B) = \max\{0, 1 - \| x \|…
user397268
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Monotonicity in an interval

Let $f:[a,b]\rightarrow R$ be differentiable at $c\in (a,b)$ with $f'(c)<0$. Does this imply that $\exists$$\delta>0$ such that f is monotonic in $(c-\delta,c+\delta)$?
Abhra Abir Kundu
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Locally uniformly convergence and differentiability of integrals

Let a function $f:[a,b)\times [c,d] \rightarrow \mathbb R$ be continuous with continuous $f_y$. Suppose that improper integral $F(y)=\int_a^b f(x,y)dx$ is convergent for $y\in [c,d]$ and integral $G(y)=\int_a^b f_y(x,y)dx$ is convergent locally…
Richard
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If $\alpha:(-1,1)\to O(n,\mathbb R)$ be a smooth map, then what can we say about $\alpha'(0)$?

Let $\alpha:(-1,1)\to O(n,\mathbb R)$ be a smooth map such that $\alpha(0)=I$, the identity matrix. Then which of the following is true? (a) $\alpha'(0)$ is symmetric. (b) $\alpha'(0)$ is skew-symmetric. (c) $\alpha'(0)$ is singular. (d)…
Mini_me
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Derivative of a sequence

Just a quick question, can you derive sequences like any normal funtion ? For example, if $a_n=\frac{1}{n}$ then $\frac{d}{dn}(a_n)=-\frac{1}{n^2}$, is this ok ?
p0ffer
  • 231
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Riemann integrable function inequality

Suppose that $f$ is Riemann integrable on $[a,b]$. Then there exists a sequence $f_{k}$ of continuous functions on $[a,b]$ so that $$\lim_{k\rightarrow \infty }\int_{a}^{b}\left | f(x)-f_k(x) \right |dx=0 $$ (My attempt) Suppose that $f$ is…
Lee
  • 441
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Two improper integrals with a parameter converge to the same function

Suppose $\int _{a}^{+\infty}f(x,y)\ \mathrm{d}y$ and $\int _{a}^{+\infty}g(x,y)\ \mathrm{d} y$ both converge to $F(x)$ on X.That is $$F(x)=\int _{a}^{+\infty}f(x,y)\ \mathrm{d}y=\int _{a}^{+\infty}g(x,y)\ \mathrm{d}y,x\in X$$ If $F(x)$ is…
Singtren
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Mathematical Analysis integral inequality

Let $f$ be a nonnegative continuous function on $[0,1]$ , and $f$ nondecrease. Then for any $0<\alpha<\beta<1$ , we have $\int_{0}^{1}f(x)dx\geq\frac{1-\alpha}{\beta-\alpha}\int_{\alpha}^{\beta}f(x)dx$ , and $\frac{1-\alpha}{\beta-\alpha}$ is the…
user56927
  • 63
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Integral. $\int_{0}^{s}{\lambda \cdot |t|^{q-2}\cdot t\text{dt}}$

How can I calculate the following integral : $$\int_{0}^{s}{\lambda \cdot |t|^{q-2}\cdot t\text{dt}}$$ $q \geq6$, $\lambda >0$ and $ s \in \mathbb{R}$. It's hard for me to calcualte this integral because I don't know what I have to do with the…
Iuli
  • 6,790
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Suppose that $f$ has the property that $f(x+y)=f(x)+f(y)$.

I am given the assumption that a function $f$ has the property that $f(x+y)=f(x)+f(y)$. I am wondering if it follows that $f(x-y)=f(x)-f(y)$. I think that this is false, yet I see this being used in proofs.
user506873
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1 answer

Is there a homeomorphism from the space of sequences to [0,1]?

If I consider the norm for the space of sequences of digits {0-9} to mimic the norm for real numbers. $|\left\{x_n\right\}| = \sum_{n=1}^{\infty} \frac{x_n}{10^n}$ shouldn't I now have a space identical to [0,1]?
user54358
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