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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for $|f (x_1) + ... + f (x_n)| \le M$ how to prove that $S=\{ x\in [0,1]:f(x)\ne 0\}$ is countable?

Let f be a real-valued function defined for every $x$ in the interval $0\le x \le 1$. Suppose there is a positive number M having the following property: for every choice of a finite number of points $x_1, x_2, ..., x_n$ in the interval $0…
S L
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If $f'''(x)$ exists on an interval $[a,b]$, does that mean $f(x)$ is continuous on $[a,b]$?

Does this follow trivially from the fact that differentiability implies continuity, and if $f'''(x)$ exists, then $f(x)$ is differentiable and therefore continuous?
Hisoka Moroh
  • 750
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Invertible idempotent in a C-star algebra question

Let $J$ be an idempotent element in a unital $C^*$ algebra. Why is $I+(J-J^*)(J^*-J)$ invertible? I have been trying to show that $\|(J-J^*)(J^*-J)\|<1$, but I could not do this.
john
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A sufficient condition for $U \subseteq \mathbb{R}^2$ such that $f(x,y) = f(x)$

I have another short question. Let $U \subseteq \mathbb{R}^2$ be open and $f: U \rightarrow \mathbb{R}$ be continuously differentiable. Also, $\partial_y f(x,y) = 0$ for all $(x,y) \in U$. I want to find a sufficient condition for $U$ such that $f$…
Huy
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How can I show this function is Weakly Sequentially Lower Semicontinuos?

Suppoe $X$ is a Banach spaces and $G\subset X$ is a convex open set. Let $\phi:G\rightarrow \mathbb{R}$ be a $C^1$ function and assume that $\phi'$ is a bounded and peseudo-monotone map (see here for a definiton of pseudo-monotone). We say that…
Tomás
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Proof of differentiability

A function $f$ is defined in $R$, and $f'(0)$ exist. Let $f(x+y)=f(x)f(y)$ then prove that $f'$ exists for all $x$ in $R$. I find this problem in here, and wonder the proof stated below has any problems? Since $f'(0)=\lim_{h \rightarrow 0}…
niagara
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Tannery's Convergence Theorem

This appears as the 7th exercise of Chapter 10 in Apostol's book of Mathematical Analysis. $\{f_n\}$ is a sequence of functions and $p_n$ is an increasing sequence such that $p_n \rightarrow +\infty$. We have that The sequence of functions…
Flashlight
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Can this function be anything but a polynomial function?

If $f\in\mathcal{C}^\infty((0,1),\mathbb{R})$ verifies the following: \begin{equation} \forall x\in(0,1),\,\exists n\in\mathbb{N},\quad f^{(n)}(x)=0 \end{equation} Can $f$ be anything but a polynomial function?
user2471
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Find the upper and lower limits of $xf(x)$, as $x\rightarrow \infty$

Define $$f(x)=\int_{x}^{x+1}\sin(t^2)dt$$ Find the upper and lower limits $xf(x)$, as $x\rightarrow \infty$. I find the answer as $+1, -1$ since $|\sin(x)| \le 1$. (Of course I calculated that function) Is that right or did I miss…
cowik
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Invariance of domain for smooth functions

Let $f \colon U \to \mathbb R^n$ ($U \subset \mathbb R^n$ open) be of class $C^1$ and injective. Apparently there is an easy proof to show that $f(U)$ is open. In general it follows from the Invariance of domain theorem. Does someone know that…
Wilhelm L.
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Proof problem explanation

Let $f$ be a function defined at $x$. Suppose that every sequence $p_1, p_2, p_3,\dots$ in the domain of $f$ converging to $x$ has the property that $f(p_1), f(p_2), f(p_3),\dots$ converges to $f(x)$. Prove that $f$ is continuous at $x$ by…
RockChalk
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Closed form solution to $\{a_n\}_{n=1}^{\infty} = 1,2,2,3,3,3,...$

I had thought about this sequence (where each positive integer $n$ shows up $n$ times) the other day and think I have a closed form solution. First of all we know that the last time that $k$ shows up in the sequence is at $a_{\frac{k(k+1)}{2}}$. We…
Patrick
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Points where function is continuous

I have a function that is defined as such, $f(x)=x$, if x is rational, ie $x=\frac{p}{q}$ and $f(x)=1-x$, if x is irrational. What are all the points of continuity? I would say that all the points of continuity are the points where $p\neq q$ since…
c-o-d
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Monotonicity of function $f(x)=(1+1/x)^x$

Possible Duplicate: How to prove $(1+1/x)^x$ is increasing when $x&gt;0$? $$f(x)=(1+1/x)^x$$ Where $x>0$ I am in search to find a proof that the function $f(x)$ is always increasing in its any real number domain. As the above function always…
Abhinav Anand
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Cosine function is decreasing on $(0,\pi)$

How to prove that cosine function is decreasing on the interval $(0,\pi)$ if we are allowed to use only the definition of cosine through exponential function (or Taylor series)?
Nasibabuba
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