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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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Theorem 1.4. of Serge Lang´s Undergraduate Analysis

Could somebody explain to me the boxed inequality. I have the feeling that the second member should be $L_1/2^{K-1}$. Thanks.
Jon Stevens
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How do I show this formula involving several variables?

This is from Woll's "Functions of Several Variables," but there's no proof. If $g$ is of class $C^k$ ($k \ge 2$) on a convex open set $U$ about $p$ in $\mathbb{R}^d$, then for each $q \in U$, $ g(q) = g(p) + \sum_{i=1}^d \frac{\partial g}{\partial…
Steven Li
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Definitions of the extended real number system: Baby Rudin vs Tom M. Apostol's _Mathematical Analysis_ 2nd edition

I have of late had the chance to go through Chapter 1 of each of the following two books: Principles of Mathematical Analysis by Walter Rudin, 3rd edition Mathematical Analysis by Tom M. Apostol, 2nd edition The first chapters of both of these…
Saaqib Mahmood
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analysis limit question

Let f be an integrable function on $\mathbb{R}$. Show that $\lim_{t\rightarrow 0} \int_{\mathbb{R}}|f(x + t) -f(x)|dx = 0$. I can make it work once it is shown to be true for $f\in C_c(\mathbb{R})$ but I am having trouble proving this case.
john
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Converging sequence and subsequences

How might we rigorously argue that if we have a sequence $\{x_n\}\subset X$ such that every subsequence of it has a convergence subsequence that tends to $a$ and $X$ is a compact set then $\{x_n\}$ converges to $a$? In my mind, I am thinking that if…
Terry
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Show that $\{x \in \mathbb{Q}:x \geq 0, x^2 \leq 2\}$ has no rational least upper bound.

Lets denote the least upper bound by $\alpha \in \mathbb{Q}$ and $\delta > 0$ be a small number. Now $\alpha^2 \neq 2$ because there is no such rational $\alpha$. If $\alpha^2 > 2$ then $(\alpha +\delta)^2 >2$ and so $\alpha$ is not a least upper…
user197848
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Upper bound for the Dirichlet kernel

I'd like to to prove the following statement: For every $N \geq 1$, there exists $C>0$ such that $|D_N(t)| \leq C|t|^{-1}$, for $|t|<\frac{1}{2}$, where $D_N(t) = \sum_{k=-N}^N e^{2\pi ikt} = \frac{\sin{((2N+1) \pi x)}}{\sin(\pi x)}$ and $\{ D_N…
Bachmaninoff
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Homeomorphism of the unit disk onto itself which does not extend to the boundary

It is well known that any conformal mapping of the unit disk onto itself extends to the unit cirle. However, is there an homeomorphism of the unit disk onto itself which does not extend to a continuous function on the closed unit disk? If yes, can…
Malik Younsi
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Proof of divergence in analysis

I aim to show that the sequence $x_n := n^2 - 10n $ diverges to $+\infty$ by using the definition of divergence (i.e. for a given $M \in \mathbb{R}$, there exists $N$ such that $n \geq N$ implies $x_n > M$). So my strategy for proving this is that…
user247618
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Proving with completeness axiom

Suppose we claim that if there is a set $E := \{a \in \mathbb{R} : a < \epsilon, \forall \epsilon \in \mathbb{Q}^{+} \}$, then it must be true that $a \leq 0$. I aim to prove this using only the ordered field axioms and the completeness axiom (as…
user247618
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Is the $n$th derivative of a continuous function also continuous?

Consider a differentiable (and hence continuous) function of order $n-1$. Is the $n$th derivative of such a function always continuous? As an example, is the $n-th$ derivative of the function $f(s) = s \exp (n-1) (s-1)$ evaluated at $s=1$ continuous…
user264948
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inequality involving $x$, $x^3$,$\sin(x),\cos(x)$

Let $x \in \left[0,\dfrac {\pi} 2 \right]$. Prove the inequality $$6x \ge 6\sin x +x^3 \cdot \cos x$$ there is nice solution using Taylor expansion. Is there other one?
Booldy
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Differentiable and $f(0)=0$

I would like to show that if $f\colon\mathbb{R}^n\to \mathbb{R}$ is differentiable and $f(0)=0$ that there exists $g_i\colon\mathbb{R}^n\to\mathbb{R}$ such that $f(x) = \sum_{i=1}^n x^i g_i(x)$. The book (Spivak) has it written like that, but I…
Squirtle
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