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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Why is the function integrable?

I was wonderind why the function $f(x)=x, x \in [0,1]$ is integrable in $[0,1]$,although $U(f,P) \neq L(f,P)$ $P$ a partition of $[0,1]$,let $P=\{ x_0=0,x_1=1\}$ $U(f,P)=(1-0) \sup f([0,1])=1$ $L(f,P=(1-0) \inf f([0,1])=0$ $\overline \int_0^1 f=1…
user159870
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Uniform distance

Uniform distance: $|f-g|_A= \sup \{ f(x)-g(x), x \in A\}, f(x)-g(x) \geq 0 \forall x \in A$ Find the uniform distance of $f(x)=x, g(x)=1 \forall x \in \mathbb{R}$ My attempt is to take cases for $x$. $x \geq 1: h(x)=x-1 \geq 0, h'(x)=1$,so $h$…
user159870
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Why is the antiderivative equal to $F(x)=|x|-1$

We have: $$f(x)=\left\{\begin{matrix} -1, & -1 \leq x \leq 0 \\ 1, & 0
evinda
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Proof of Cauchy's functional equation for rational arguments

We have thesis that for every $c\in\mathbb{Q}$ every additive function has form of $f(x)=cx$. In the proof we're showing that $f(nx)=nf(x)$. Then we're supposed to replace $nx$ by $\frac{1}{n}x$. Why can we do this?
guest
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analysis, uniform continuouty

Have this in one of my finals practice questions: Given $f(x) = e^{\sin x}$, prove $f(x)$ is uniformly continuous on $\mathbb R$. The direction I'm thinking of is choosing $x_1 = 2\pi k + \delta$, and $x_2 = 2\pi k + \frac{\delta}2$. so I need to…
user2970357
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A question regarding the inverse of continuous mappings.

If there is a continuous mapping $f:\Bbb{R^2}\to\Bbb{R}$, will $f^{-1}$ also be continuous? If there is a differentiable mapping $g:\Bbb{R^2}\to\Bbb{R}$, will $g^{-1}$ be continuous/differentiable? I don't know how to proceed here. To show that…
freebird
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Infinitely small functions!

Suppose that $\lim_{x\to a} f(x) =A$ and $\lim_{x\to a} g(x) =B$ and $B \neq 0$. Then $\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)} = \frac{A}{B}$. This is really well known and very easy to prove by the regular…
davd
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A function that converges monotonically?

Let $f_n(x):=n\left(e^{\frac{x^2}n}-1\right)$ on $\left[0,M\right]$. I believe this converges monotonically to $x^2$. I used L'Hopital's rule to show pointwise convergence and the ratio of the derivatives I got converges monotonically to $x^2$…
user9352
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Rough estimate for the fourier coefficients of a discrete bump function

Setup: Let $p$ be a large prime number and $F = \{0, \dots, p-1\}$ be the field of order $p$. Let $I$ denote the discrete interval $I = \{1, \dots, M\}$ for some $M < p$. Regard both $I$ and $F$ as subsets of the real line, and let $f : [0,p) \to R$…
JMD
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Find flux through $S=\{z=2(1-x^2-y^2),\ z\geqslant0\}$

Let $S=\{z=2(1-x^2-y^2),\ z\geqslant0\}$. How to calculate flux of $rot\ A$ outside $A$ through $S$ for $A=(y,z,x)$ ? I know the definition, but don't know how to actually calculate what's mentioned in this exercise...
Jules
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Remains true for f integrable?

$f(x)=f(x+1)\ \forall x\Rightarrow \int_0^1f(x+t)dt=\int_0^1f(t)dt$, when $f$ is continuous on $[0,1]$. The proof it is not hard. My question is, this property remains true if $f$ is only integrable?
bob
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To write a relation for $g \circ f$,do I have to pick $\epsilon=\delta'$?

I am looking at this exercise: Let $f:A \to B, g: B \to \mathbb{R}$ If $f$ is uniformly continuous on $A$ and $g$ is uniformly continous on $B$,show that $g \circ f$ is uniformly continuous. $f$ is uniformly continuous on $A$: $\forall \epsilon>0 \…
evinda
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how do we get this inequality: $\frac{|x-y|}{\sqrt{x}+\sqrt{y}} \leq \frac{1}{2}|x-y|$?

To show that $f(x)=\sqrt{x}$ is Lipshitz continuous,we do it like that: $$|f(x)-f(y)|=|\sqrt{x}-\sqrt{y}|=\frac{|x-y|}{\sqrt{x}+\sqrt{y}} \leq \frac{1}{2}|x-y|$$ But...how do we get this inequality: $\frac{|x-y|}{\sqrt{x}+\sqrt{y}} \leq…
evinda
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Are these roots the only ones or are there more roots?

I am looking at the following exercise: Let $f:[a,b] \to \mathbb{R}$ integrable. Show that $\exists \xi \in [a,b]$ such that $\int_a^{\xi} f(u) du= \int_{\xi}^{b} f(u) du$ We define $G(x)=\int_a^{x} f(u) du+ \int_{b}^{x} f(u) du$ $$\int_a^{x} f(u)…
evinda
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How to prove a limit?

I saw at the solution of an exercise that $$\frac{|x|^{n+1}}{(n+1)!} \to 0, \text{ when n } \to +\infty$$ But,how can I show that it is actually like that?
evinda
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