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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How to prove that $\exp(x)$ and $\log(x)$ are inverse?

How does one prove that the exponential and logarithmic functions are inverse using the definitions: $$e^x= \sum_{i=0}^{\infty} \frac{x^i}{i!}$$ and $$\log(x)=\int_{1}^{x}\frac{1}{t}dt$$ My naive approach (sort of ignoring issues of convergence) is…
user140776
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$f$ is monotonically increasing, $0 \le f \le 1$ and $\int_0^1 (f(x) - x) dx = 0$ then $\int_0^1|f(x)-x|dx \le \frac{1}{2}$.

$f(x)$ is monotonically increasing in $[0,1]$, $0 \le f \le 1$ and $\int_0^1 (f(x) - x) \mathrm{d}x = 0$. Prove that $\int_0^1|f(x)-x|\mathrm{d}x \le \frac{1}{2}$. It's easy if $f(x) \ge x$ in $[0,1]$. And even in $[a,b]$ we have $\int_a^b…
XT Chen
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Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B $ is not closed

Possible Duplicate: Sum of two closed sets in $\mathbb R$ is closed? Give an example of two closed sets $A, B \subseteq \mathbb{R}$ such that the set $A + B = \{a + b : a \in A, b \in B\}$ is not closed. This question appears on an old analysis…
JZS
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Is $\lim_{n \to \infty} \lim_{l \to \infty} \ a_{n,l} = \lim_{l \to \infty} \lim_{n \to \infty} \ a_{n,l}\;$?

Is $\lim_{n \to \infty} \lim_{l \to \infty} \ a_{n,l} = \lim_{l \to \infty} \lim_{n \to \infty} \ a_{n,l}\;$? What happens if I replace limits with lim sups? Thanks!
UH1
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will a nonempty countable and compact subset of a metric space always contain an isolated point?

Let $(X,d)$ be a metric space, $K \subseteq X$ is nonempty, countable, and compact. I could not come up with an example where such a K has no isolated point(s). so I want to prove that an isolated point will always exist. I tried by first assuming…
Frakakka Feet
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How to prove $\max_{x \in I} |f(x)| \leq \max_{x \in I} |f'(x)|$?

Today we had a probational exam in analysis. I wasn't able to solve one of the exercises and I have no idea what theorem to apply in order to solve it: Let $I=[0,1]$ and $f: I \rightarrow \mathbb{R}$ be continuously differentiable. Assuming that $f$…
Huy
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Constructing a larger rational whose square is also less than two

Possible Duplicate: Rational Numbers Baby Rudin has a very nice construction showing that, given a positive rational number whose square is less than (greater than) two, one can always find a larger (smaller) rational whose square is also less…
David
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Minimum of $x^2+\frac{a}{x}$ without Calculus

How can I find the minimum of $x^2+\frac{a}{x}$ on $\mathbb{R}_+$ without calculus?
user6703
8
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4 answers

Composition of functions that are onto or one-to-one

I found part of my answer here: If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that...); however I wanted to flesh out the last two statements I had in a proposition in my notes. Proposition: Let $f: A \rightarrow B$…
emka
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$f$ strictly increasing does not imply $f'>0$

We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'>0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any counterexample that shows the converse fails? I have been…
Daniel
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What is a formal definition of series?

Is there a formal definition for series? For example, cardinal sum has a formal definition such that $\sum a_i$ = $\bigcup a_i$. Is there any clear definition for series of real or complex number? The definition on my book is the sum of $a_0 + ... +…
Katlus
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Wick Rotation technique

I am trying to get my head around the Wick rotation technique. I have tried to play around with some elementary examples. Let us imagine I need to solve on the real line $$ y’ = \cos (x)$$ the prime denoting differentiation. I can view the function…
An aedonist
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Does for every continuous function $f:R \rightarrow R$ there exist a sequence of analytic functions convergent uniformly to $f$?

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called $\mathbb{R}$-analytic iff for every $x_0 \in \mathbb{R} $ there exist $R>0$ and power series $\sum_{n=0}^\infty a_n (x-x_0)^n$ convergent for $|x-x_0|
A.B
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How find the example such $\left(\sup_{x\in R}|f'(x)|\right)^2=2\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$

Question: Find a example function $f$,such $f\in C^2(R)$,and such $$\left(\sup_{x\in R}|f'(x)|\right)^2=2\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$$ This problem is from when I prove this inequality $$\left(\sup_{x\in R}|f'(x)|\right)^2\le…
math110
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Is there a pair of open sets whose Minkowski sum is not open?

Is it possible to find an example where the Minkowski sum of two open sets is not open? (If someone could think of one, could they possibly also suggest how they came up with the example? Perhaps there is a "common counterexamples" list that people…
bart
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