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Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

Usually includes topics related to measure theory, integration theory and functional analysis.

5277 questions
50
votes
5 answers

Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable? I ask for pedagogical reasons. Results in basic real analysis relating a function and its…
Mark Meckes
  • 11,286
41
votes
2 answers

Square root of a positive $C^\infty$ function.

Suppose $f$ is a $C^\infty$ function from the reals to the reals that is never negative. Does it have a $C^\infty$ square root? Clearly the only problem points are those at which $f$ vanishes.
Michael Barr
  • 887
36
votes
12 answers

Examples where existence is harder than evaluation

In expressions involving an infinite process (infinite sum, infinite sequence of nested radicals), sometimes the hardest part is proving the existence of a well-defined value. Consider, for example, Ramanujan's infinite nested radical: $$…
Aryeh Kontorovich
  • 6,061
23
votes
3 answers

Is there a function defined on real numbers which is continuous from the left, but not from the right, everywhere

I am teaching Mathematical analysis. A student asked this question. I think this is a good question, but don't know the answer.
Hao Yin
  • 527
20
votes
1 answer

Can all partial sums $\sum_{k=1}^n f(ka)$ where $f(x)=\log|2\sin(x/2)|$ be non-negative?

Let $f(x)=\log|2\sin(x/2)|$ (the normalizing factor $2$ is chosen to have the average over the period equal to $0$). Does there exist $a>0$ such that all sums $\sum_{k=1}^n f(ak)\ge 0$? The computations (run up to the values of $n$ where I could not…
fedja
  • 59,730
20
votes
1 answer

Is every function $f: \mathbb R \to \mathbb R$ differentiable at at least one point when restricted to some everywhere dense subset of $\mathbb R$?

I was doing some fairly simple research a few hours ago and I almost asked a similar question with the word continuous instead of differentiable in the title, but then I found this question asked by Gro-Tsen where there is an affirmative answer to…
user153451
20
votes
0 answers

Is $\sum_{n=1}^\infty \frac{n!}{n^n}$ rational?

Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n}$ rational? This question has been posted in MSE for two years without an answer. A094082 seems to suggest that it is not rational. Is it still an open problem? Are there any known results about…
user14319
19
votes
3 answers

functions from Q to itself with derivative zero

Let $f: {\bf Q} \rightarrow {\bf Q}$ be a "${\bf Q}$-differentiable" function whose "${\bf Q}$-derivative" is constantly zero; that is, for all $x \in {\bf Q}$ and all $\epsilon > 0$ in ${\bf Q}$, there exists $\delta > 0$ in ${\bf Q}$ such that for…
James Propp
  • 19,363
18
votes
2 answers

Is there an analytic non-linear function that maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers?

Consider a function $h$ defined on real numbers, which is not of the form $kx+b$ i.e. a linear function. If $h$ maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers, could $h$ be analytic? If so, how to give…
Frank
  • 193
17
votes
1 answer

Continuous functions of three variables as superpositions of two variable functions

Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's work on this problem, but it seems that in that…
KhashF
  • 2,608
16
votes
3 answers

Which functions have all derivatives everywhere positive?

Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$. The only examples I can construct are the functions $ae^{bx}+c$ for…
Will Sawin
  • 135,926
16
votes
1 answer

Are continuous functions almost completely determined by their modulus of continuity?

Given a function $f: \mathbb{R}\to\mathbb{R}$, we define its left modulus of continuity, $L(f): \mathbb{R} \times (0, \infty)\to [0,\infty]$ by $$L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)) \subseteq [f(x) - e, f(x) + e]\} $$ Similarly define the…
James Baxter
  • 2,029
15
votes
1 answer

Solving a non linear equation

I've been trying to prove that the following equation has a unique solution in interval 0 < x < 1 : $$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$ Where K is a number (integer, if it helps) greater than 1. I have checked it numerically and,…
freedome321
  • 159
14
votes
2 answers

Generalisation of Cauchy's mean value theorem

I apologise in advance if this is an elementary question more fitted for Math Stack Exchange. The reason why I have decided to post here is that the question I am used to seeing on that site are not of the open-ended format of the one I am…
carfog
  • 413
13
votes
1 answer

How continuous can a bijection between line and plane be?

Is there a bijection $f$ from $[0, 1]$ to $[0, 1]^2$ such that the set of points of discontinuity of $f$ has measure zero? If not, could it be dense/comeager?
Milo
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