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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Real Analysis Delta Epsilon proof

Suppose $f: D \to \mathbb{R}$ and $x_0$ is a limit point of $D$. Prove that $\lim_{x\to x_0} = L$ if and only if for every $\varepsilon>0$, there exists a $\delta>0$ such that if $x$ is in $D$ and $0 <|x - x_0|<\delta$ then $|f(x) - L| <…
RSt
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Interchanging the limits,

so I really suck at analysis and I want to get better doing some problem on my own. I encountered this one today: So I was given the following problem: $f:(a,b)\rightarrow \mathbb{R}$ be a continuous and differentiable in $(a,b)\backslash\{c\}$. If…
Daniel Montealegre
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Sketching subsets of R2

The assignment reads sketch the following subsets of $\mathbb{R}^2$: (a) $\{(x, y) \in \mathbb{R}^2 \mid x > [y]\}$ (b) $\{(x, y) \in \mathbb{R}^2 \mid |x|^p + |y|^p < 1\}~\mathrm{for}~p = \frac{1}{2}, 1, 2, 4$ (c) $\{(x, y) \in \mathbb{R}^2 \mid…
phil demeanour
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Prove or Disprove: the product of the upper bounds is an upper bound of the Minkowski product

Prove or find a counterexample to the following statement: If $A$, $B \subseteq \mathbb{R}$ are nonempty, $M$ is an upper bound for $A$ and $N$ is an upper bound for $B$, then $MN$ is an upper bound for $AB := \{ab \mid a \in A, b \in B\}$ Now…
phil demeanour
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Mathematical Analysis: Mean Value Theorem

I'm trying to learn how the mean value theorem works by attempting to answer the following but i just dont understand it at all. Consider the function $f(x)$ $=$ $\sqrt x$ for $x > 0$ i. Show that $f'(x)$ is a decreasing function on $(0,…
Euden
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Prove compact sublevel sets imply coercivity

$f$ is convex and $dom(f):x \in \mathbb{R}^N$. Define sublevel sets of $f$ as \begin{equation} \mathbf{S}(f,\beta)=\{x \in \mathbb{R}^N\ : f(x) \leq \beta \} \end{equation} are compact. I need to prove that if $f$ is a convex function, then having…
NAASI
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$C[0,1]$ and two of it's subsets, are they dense in the vector space or not.

I am not sure of my answers to this question. I would appreciate if someone can verify if my logic is correct, Thanks! So $C[0,1]$ is the vector space of all continuous function on the interval [0,1]. (over the real numbers) with the norm $||f||=…
Charles Carmichael
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Is there any counter-example to $\inf_{x, y}f(x,y) = \inf_{x}\inf_{y}f(x,y)?$

Thanks! I couldn't think of any counter-examples.
Yuan Gao
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A nonempty compact convex subset $A\subset \mathbb{R}^n$ has an extreme point.

A nonempty compact convex subset $A\subset \mathbb{R}^n$ has an extreme point. How do you prove this result? Can you give me sketch? Thanks
md_bar
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Linear Isomorphism And Completeness

I am just a beginner. So please be patient with me. Consider $X=\{\frac 1 n : n \in \mathbb N\} \cup \{0\}$. Prove that the vector space of continuous functions on $X$ is linearly isomorphic to the space of convergent sequences in $\mathbb R$.…
Ester
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Explanation of uniqueness of square root

Let $c$ be a positive number. Then there is a unique positive number whose square is $c$. That is, $x^2=c$ Start: Suppose $a$ and $b$ are numbers whose square is $c$. then $a^2=c$ and $b^2=c$ $c-c=0 = a^2-b^2 = (a-b)(a+b)$ We know $(a+b) > 0$…
user11460
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A mapping dominates another

I was wondering what the definitions of one mapping dominating another in some general settings are? A special case I inferred from Dominated Convergence Theorem is that: for mappings $f$ and $g$ from a set $X$ to $\mathbb{R}$, $f$ is called to…
Tim
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Demonstrate by induction

Let $p,q \in \mathbb{N}$, be prime among them and $p
hlapointe
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Indicator function defined on the empty set

Let $G \subset X$ be sets and $I_{G}$ be an indicator function, i.e. $I_{G}(x) := 1$ if $x \in G$ and $I_{G}(x) := 0$ if $x \notin G.$ But does the above definition cover the case where $G$ is empty?
Yes
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How to evaluate $\int \limits_{-\infty}^{\infty}\frac{e^{-|x|}}{|1-\sin x|^{\frac{1}{4}}} \,dx$?

$$\int \limits_{-\infty}^{\infty}\frac{e^{-|x|}}{|1-\sin x|^{\frac{1}{4}}} \,dx$$ Any advice and comments will be appreciated
yaoxiao
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