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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What is the maximum of a function over the empty set?

could you answer my question? Let $A \subset \mathbb{R}$ be a subset of the real numbers and $f$ a continuous functions. What happens the function $\sup_{t \in A} |f(t)|$ if $A$ is chosen to be the empty set? I feel like it is undefined but can…
PROB123
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Proving a sum divergent

Hello I am solving the following problem. Define a sequence of real numbers $(a_n)$ as follows: Let $a_1$ be any real number satisfying $0 < a_1 < 1$, and define $a_2, a_3,\dots$ recursively via $a_{n+1} := \cos(a_n)$ . Prove the series $\sum_n…
Lavorizia Vaughn
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Limit of $\frac{f(x)}x$ for some real function $f$

Let $f$ be a differentiable function in $[1,\infty) \to \mathbb R$ with a continuous derivative for which $$ f'(x) = \frac 1{x+2010f(x)^2} , \quad x\ge 1 $$ and $f(1)=0$. Find $\lim_{x\to\infty} f(x)/x$.
Plom
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Determine whether the following series converge or diverge for the values of p.

I need some real help with this question. I have tried this question already and I am so stressed out from it. So what I did was that I used the Ratio Test but I couldn't get the result.
Rabia K.
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Proving that a function $f$ is a contraction

Hello I am solving the following problem and could use some help. Let (C[0,1],$d_\infty$) be the metric space of continuous functions on [0,1] where the distance function is defined by Let $d_\infty(f,g)=\sup_{x∈[0,1]}|f(x)−g(x)|. $ Consider the…
Lavorizia Vaughn
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How should I continue my mathematical studies after reading Michael Spivak’s calculus?

I have recently finished calculus by Michael Spivak, and wanted to learn more about things like complex analysis talked about towards the end of this book.I was also very interested in learning more about concepts like analytic continuation. Which…
A.G
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Show the derivative exists for any x $\in \mathbb{R}$, but $f' : \mathbb{R} \to \mathbb{R}$ is not a continuous function.

I am solving the following problem: I am to consider a function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x)=x^2sin(\frac{1}{x})$ when $x \neq0$ and $f(x)=0$ when $x=0$. And I am to show that the derivative f'(x) exists for any x $\in…
Lavorizia Vaughn
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How prove $\mathbb Q$ is close in the following metric space?

assume $(d,\mathbb R)$ be a mertic space such that $$d:\mathbb R\times \mathbb R \to [0,\infty)$$$$d(x,y)= \begin{cases} 0, & \text{if x=y} \\ max\{|x|,|y|\}, & \text{if x$\neq$y} \\ \end{cases}$$ How prove $\mathbb Q$ is close in this metric…
M.H
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differentiation of a volume

Given a function $f(x)$. Then we define $D=\{x\mid f(x)>y\}\subset\mathbb{R}^n$ and $\Gamma=\{x\mid f(x)=y\}$. Now we define $S(y)=\int_D dx$. My question is what is the meaning of $S(y)$? Is it "size" (volume) of the domain $D$? How can I obtain…
m15
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Prove that the given function: $h(x) = x^2$ is continuous at every real number

Use the definition of continuity to prove that the given function: $h(x) = x^2$ is continuous at every real numbers. So far for my proof I have: Let $\epsilon >0$ be given to us. We must show there exists a $\delta >0$ such that $$|x-c| < \delta…
Yogibear
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Continuity of function with "internal" functions

Suppose we have some $h:\mathbb{R}\to\mathbb{R}^2$ where $h(x)=(f(x),g(x))$ where $f,g:\mathbb{R}\to\mathbb{R}$. $h$ is continuous $\iff f,g$ are continuous. I'm not sure exactly what these sorts of functions are called, as they're not exactly…
Adam Zapel
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Absolute value proof in Spivak

Prove (1) using the fact that $|x|-|y| \le |x-y|$ (1) $|(|x|-|y|)| \le |x-y|$ Attempt: as I know $|a| \le b \iff -b \le a \le b$, then proving (2) would allow me to prove (1) (2): $-|x-y| \le |x| - |y| \le |x-y|$. From testing different cases of $x$…
Eliot Behr
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Proving that B(X) is a Metric Space

I was hoping for help with the following problem. Let X be the set of all bounded functions existing in B(X). For $f_1,f_2∈B(X)$ I am to show that $d({f_1}, {f_2}) = sup{[f_1(x) − f_2(x)| : x∈X, n ∈ N}$ is a metric space. I know that all bounded…
finkleberrys
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How to make an injective function $ f $ such that $ f : \left[0,1\right]\rightarrow2^\mathbb{N} $ using binary notation?

$ 2^\mathbb{N} $ is a power set of $ \mathbb{N} $. By using binary notation, we can take $ x\in\left[0,1\right]$ and $ x=0.a_1a_2a_3\cdots\left(a_n\in\left\{0,1\right\}\right) $ Then we can make a set $ A=f(x) $ as an element of $ 2^\mathbb{N} $…
Park Inseo
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Berkeley Problem 1.1.6

The book gives a more complicated proof than I wrote, I was wondering if mine was incorrect: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be nonconstant and assume $x \leq y$ implies $f(x) \leq f(y)$. Prove that there exist $a \in \mathbb{R}$ and $c…
Mikolaj
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