# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

89,005 questions
0answers
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### Average value of the sum $\sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \{ \log_{b} a^{-k} x \}$

Let $a$ and $b$ be positive integers both greater than $1$. I'd like to compute the average value of the summation \begin{eqnarray} \sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \left \lbrace \log_{b} a^{...
2answers
740 views

### On differentiability and definition of derivative in case of uniform convergence

Consider a sequence of functions $f_n$ where $f_n : \mathbb{R} \to \mathbb{R}$ and $f_n$ are all differentiable with derivatives $f^\prime_n$. The sequence $f_n$ and the sequence $f^\prime_n$ both ...
5answers
19k views

### what functions or classes of functions are $L^1$ but not $L^2$

We know that if a real function is in $L^2$ then it is in $L^1$, but the reverse is not necessarily true. So what are the examples of functions that are $L^1$ but not $L^2$, especially those ...
3answers
1k views

### what functions or classes of functions are Riemann non-integrable but Lebesgue integrable

I am wondering if there are some other examples of Riemann non-integrable but Lebesgue integrable, besides the well-known Dirichlet function. Thanks.
2answers
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1answer
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### Is there a compactly supported smooth function which is exactly k times differentiable at exactly one point?

Is there a compactly supported and smooth (except at one point) function $f : \mathbb{R} \to \mathbb{R}$ where (non smooth point) it is is exactly k times differentiable ?
1answer
387 views

### One-sided limits of a uniformly convergent function sequence and its limit function

I'm having some difficulty with the following question: Let $(f_{n}(x))$ be a uniformly convergent functions sequence in $(a,b)$ (where b can be $\infty$) such that $(f_{n}(x)) \to f(x)$. Suppose ...
1answer
195 views

3answers
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### Average value of the Fractional Part of a $C^{k}$-function

The mean value of the fractional part $\{ x \}$ on $\mathbb{R}_+$ is clearly $\tfrac{1}{2}$. I'd like to know if a similar statement holds for $\{ f(x) \}$ where $f \colon X \to \mathbb{R}_{+}$ is a ...
1answer
103 views

### If $p\in\mathbb{Q}^+$, for any $x\in\mathbb{R}$, there is a rational $q\in x$ such that $p+q\not\in x$

I just read about the construction of the real numbers in Enderton's Elements of Set Theory, and am now trying to go through all the exercises. Enderton chooses to construct the reals with left sides ...
2answers
828 views

### Chicken-Egg problem with Fubini’s theorem

Fubini's theorem states that if you have a double integral over a function $f(x,y)$, then you can compute the integral as an iterated integral, if $f(x,y)$ is in $\mathcal{L}^1$. But to find out if $f$...
3answers
922 views

### If $g\geq2$ is an integer, then $\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}}$ and $\sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$ are irrational

How do we show that if $g \geq 2$ is an integer, then the two series $$\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} \quad \ \text{and} \ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$$ both converge ...
1answer
288 views

### “On the consequences of an exact de Bruijn Function”, or “If Ramanujan had more time…”

In this question on Math.SE, I asked about Ramanujan's (ridiculously close) approximation for counting the number of 3-smooth integers less than or equal to a given positive integer $N$, namely, \...
3answers
671 views

### Set Theory as it Relates to Number Systems?

I've been referred to this website, hopefully you have the background in set theory to help me out here. Got two questions, the first is on number systems arising out of set theory and the second ...
1answer
616 views

### Formula of angle between longest diagonal and any edge of a cube

Let $a<b$ be real numbers. Define a cube $Q$ in $R^n$ to be the $n$-fold Cartesian product of $[a,b]$ with itself. Find a formula of the angle between the longest diagonal of $Q$ and any of its ...
1answer
135 views

### Continuum limit of (DE) equation

I have a system described an equation, and I want to find an (DE) equation for z(x,t), in the limit as l->0. First some definitions to simplify it some: $Z1=z(x,t)-z(x+l,t)$ $Z2=z(x,t)-z(x-l,t)$ ...
2answers
6k views

### Monotone+continuous but not differentiable

Is there a continuous and monotone function that's nowhere differentiable ?
1answer
625 views

### Differentiable+Not monotone

Is there a real function that is differentiable at any point but nowhere monotone?
2answers
2k views

1answer
132 views

### How $\log(t)$ can be the limit of $t^r$, where $r\to 0$?

My class is solving the Cauchy-Euler differential equation $a t^2 y'' + b t y' + c y = 0$. The solutions are powers of $t$, $y = t^r$ and then you solve for $r$ using the characteristic ...
2answers
924 views

### Is the the sum of two locally bounded functions also locally bounded?

Does someone know if the function resulting of the sum of two locally bounded functions is also locally bounded? Thanks in advance!
1answer
173 views

### Proving the partial derivative of a strictly decreasing function is finite everywhere

Suppose I have a function $F:[0,1] \times [0,\infty) \rightarrow (0,1]$ of two variables $(s,t)$ that satisfies: (1) $F(s,0)=1$ for each $s \in [0,1]$. (2) $\lim_{t \rightarrow \infty} F(s,t)=0$ for ...
4answers
832 views

3answers
3k views

### Norms on C[0, 1] inducing the same topology as the sup norm

This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying. Let $C[0, 1]$ denote the ...
2answers
824 views

### Corollary to Baire's Category Theorem

In Rudin's Real and Complex Analysis (p. 97 in my 3rd edition), the following is stated as a corollary to Baire's Category Theorem: "In a complete metric space, the intersection of any countable ...
2answers
94 views

### How to show that $f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$

How to show that: $$f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$$ is the Dirichlet's function.