# Questions tagged [real-analysis]

For questions about real analysis, such as limits, convergence of sequences, properties of the real numbers, the least upper bound property, and related analysis topics such as continuity, differentiation, and integration.

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### Decide if f is Riemann integrable on [0,1] if the answer is affirmative, compute f(t)dt. See image attached.

Here is my progress, but not sure if the double summation is correct.
1 vote
35 views

### Is every continuous bijection from $\mathbb{R}$ to itself of finite even order an involution?

The following question is inspired by Michael Penn's YouTube video Functions that "cube" to one.. Given any continuous bijection $f:\mathbb{R} \to \mathbb{R}$ for which the order of $f$ in ...
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### Can you critique my exposition of $\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt \pi$?

Prove $$\int_{-\infty}^{+\infty}e^{-x^2}dx = \sqrt \pi$$ using Fubini's Theorem. My solution is below. Proof is Correct. What I want to know is : Is it well written? How could the writing be improved, ...
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### For which values of $\alpha \in \mathbb{R}$ does $\sum_{n=1}^{\infty}n^\alpha \left (\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}} \right )$ converge

I have a question in my analysis course that I have been trying for a long time and I'm not quite sure how to answer it. I have an idea for an answer but I'm not sure if it's right. Also note the only ...
1 vote
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### Differentiable function satisfying $x \cdot \nabla f(x) = 0$

We have a differentiable function $f: \mathbb{R}^n \setminus \{0\} \rightarrow \mathbb{R}$ that satisfies $$x \cdot \nabla f(x) = 0 \quad \forall x \in \mathbb{R}^n \setminus \{0\}$$ I am trying to ...
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### Requirements for convexity of isolated minima

Let $\mathcal{F} \in \mathscr{C}^1(X, \mathbb{R}^+ \cup {0})$, where $X$ is a separable Hilbert space. Consider any isolated minimum $\phi \in X$ ($\phi$ has a non-zero neighborhood in which it is the ...
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### How to prove a result about the accumulation point / cluster point in Brownian Motion?

I have a standard Brownian motion, $B(t)$, $t\ge0$. I am trying to prove the following result: Every $t > 0$ is an accumulation point (i. e., cluster point) of $(s: B(s) = B(t))$ from the right, ...
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### convergence of a numerical series using information about an entire series

I'm on a problem that seems simple but turns out to be a bit twisted. Let be $\sum_{n\epsilon N }^{}{u_nz^n}$ a power series with radius of convergence ρ = 1. Which of the following statements are ... 1 vote
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### Area of a surface by means of a double integral

Considering the surface in $\mathbb{R^3}$ given by: $x^2-y^2=1$; $x>0;-1<y<1;0\leq z \leq 1$ Calculate its area by a double integral via a parametrization of the surface. Firsly I setted ...
1 vote
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### $g(\vec{x}) = max f_1, \ldots , f_m \text{, with } f_k \text{ continuous at } \vec{a} \text{ where } 1 \le k \le m$ Is $g$ continuous at $\vec{a}$?

$$g(\vec{\mathbf{x}}) = max f_1, \ldots , f_m \text{, with } f_k \text{ continuous at } \vec{\mathbf{a}} \text{ , where } 1 \le k \le m. \\\text{ Discuss the continuity of } g(\vec{\mathbf{a}}).$$ A ...
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### Disproving uniform continuity of a function using Cauchy continuity

From my understanding of uniformly continuous functions, they will, by definition, map Cauchy sequences to Cauchy sequences (thus preserving the Cauchy sequence in its transformation). If a function ...
1 vote
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### For $x\in[0,1]$ find $\lim_{n\to\infty}\frac{n^2 x}{e^{n^2 x^2}}$

now this expression gives infinity over infinity so l'hospital rule can be used and the limit is zero which makes sense because the exponential is faster . but my question is what happens when x is ...
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### What does the notation $D$ mean here?

So what does the notation $D$ mean in the definition of relative entropy? I try to explain the $D$ as $\nabla_{(\rho,u)}$, but it seems not right. The article is "EXISTENCE AND UNIQUENESS OF ...
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### Use sequential continuity to show that f = g everywhere [closed]

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ are continuous functions, and that we have $f(x) = g(x) \ \forall \ x \in \mathbb{Q}$. Use sequential ...
53 views

### A bounded function, which has a primitive function, that is not continuous in more that a finite number of points?

I am looking for an example of a function that is bounded on a compact interval and has a primitive function, but that is not continuous on that same interval in more than just a finite number of ...
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### A reference for the Weierstrass epsilon-delta definition of continuity

I am trying to locate a good reference related to the ${\varepsilon-\delta}$ definition of continuity. In particular, according to Wikipedia, the definition was stated respectively by Weierstrass and ...
1 vote
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### Question regarding integration by substitution and Riemann sums

I was reading Advanced Calculus: A Geometric View and I liked how he explained in integration by substitution why the differential changes the way it does(both in push-forward and pullback ...
42 views

### Stochastic process vs. random function

On the wiki page on stochastic processes, it says that they can be interpreted as random elements of a function space, which makes complete sense to me. However, it goes on to say that this ...
### A doubt about the definition of absolute continuity on $\mathbb{R}$.
We say tha a collection of closed intervals in $\mathbb{R}$ is non-overlapping if their interiors are disjoint. A real- valued function $f$ on a finite closed interval $[a,b]$ is said to ve absolutely ...