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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Does there exist $f:(0,\infty)\to(0,\infty)$ such that $f'=f^{-1}$?

Recently the following question was posed: does there exist a differentiable bijection $f:\mathbb R\to\mathbb R$ such that $f'=f^{-1}$? (Here, $f^{-1}$ is the inverse of $f$ with respect to composition of functions.) The answer, as it turns out, is…
Dejan Govc
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The preimage of continuous function on a closed set is closed.

My proof is very different from my reference, hence I am wondering is I got this right? Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of continuous function on closed set is closed. Let $D$…
1LiterTears
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Sum of two periodic functions

Let $f$ and $g$ be two periodic functions over $\Bbb{R}$ with the following property: If $T$ is a period of $f$, and $S$ is a period of $g$, then $T/S$ is irrational. Conjecture: $f+g$ is not periodic. Could you give a proof or a counter example? …
AgCl
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Taylor expansion at infinity

Are there case where does make sense to speak about the "Taylor expansion of a function ad infinity"? By inversion, sending $x \to \frac{1}{x}$ one could exchange $0\leftrightarrows\infty$; then if the values of the derivatives of a function are…
Immanuel Weihnachten
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Why is the rational number system inadequate for analysis?

In the very first chapter of Principles of Mathematical Analysis, the author pointed out as follows: The rational number system is inadequate for many purposes, both as a field and as an ordered set. For instance, there is no rational $p$ such that…
eca2ed291a2f572f66f4a5fcf57511
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How to prove uniform continuity?

I'm starting out university math and I'm struggling with understanding how to prove uniform continuity. I think I understand the concept of finding a $|x-x_0|<\delta$ for $|f(x)-f(x_0)|<\epsilon$ but all the examples I have found so far have been…
supset
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How to prove there exists $c$ such $f(c)f'(c)+f''(c)=0$

Nice Question: let $f(x)$ have two derivative on $[0,1]$,and such $$f(0)=2,f'(0)=-2,f(1)=1$$ show that: there exist $c\in(0,1)$,such $$f(c)f'(c)+f''(c)=0$$ my try: since $$f(0)=2,f'(0)=-2,f(1)=1$$ so we easy $$f(x)=x^2-2x+2$$ such this…
user94270
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4 answers

Are angles ever multiplied?

I recently explained multiplication of (non-zero) complex numbers to my Mathematics Fundamentals students, the usual bit about “multiply their lengths, and add their angles”. Of course, there is always at least one student who wants to know why the…
Mike Jones
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Swapping signs in analysis proofs

Under what minimal conditions are the following interchange of operations valid (including a question of existence, if not given explicitly)? \begin{align*} \lim \int f_n&=\int \lim f_n \\ \lim_{x\to a} \lim_{y \to b} f(x,y)&=\lim_{y\to b}\lim_{x…
Darrin
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What are all measurable maps $f:\mathbb C\to\mathbb C$ such that $f(ab)=f(a)f(b)$?

Is there a nice description of all nonzero measurable functions $f:\mathbb{C}\to\mathbb{C}$ such that $f(ab)=f(a)f(b)\ $ for all $a$ and $b$ in $\mathbb{C}$? This is inspired by the question Multiplicative Analytic Functions, Theo Buehler's…
Jonas Meyer
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6 answers

Must a monotone function have a monotone derivative?

If a function is differentiable and monotone on the interval $(a, b)$, then its derivative is also monotone on $(a, b)$. How do you prove this statement is wrong? Can you please provide an example?
resha
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Is there a number that's right in the middle of this interval $(0, 1)$?

This might seem like a silly question, but is there a number that's right in the middle of this interval $(0, 1)$? And the half-open intervals: $(0, 1]$, $[0, 1)$? I know for a fully closed interval $[0, 1]$ it's $1/2$ because that's half the length…
Mark
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What is the point of countable vs. uncountable sets?

I understand how to use these concepts and how to prove certain sets are countable or uncountable. However I don't get the point of it. What difference does it make whether a set is countable? People say that Cantor's proof that the real numbers are…
user75122
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Proving that a convex function is locally Lipschitz

I am trying to show that if $f$ is convex in $(a,b)$ it is Lipschitz in $[c,d]$ where $a \lt c \lt d \lt b$. Here's what I have so far: Let $t_1,t_2 \in \mathbb{R}$ such that $a \lt t_2 \lt c \lt d \lt t_1 \lt b$ and let $x_1,x_2 \in…
yotamoo
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5 answers

What is the difference between a supremum and maximum; and also between the infimum and minimum?

What is the difference between a supremum and maximum; and similarly the infimum and minimum? Also, how does one tell if they exist? Here is an example: $$x_n = \frac{n}{2n-1}$$ Determine whether the maximum, the minimum, the supremum, and the …
Stefan Vrecic
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