Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Real analysis is a branch of mathematical analysis, which deals with real numbers and real-valued functions. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the limits of sequences of functions of real numbers, continuity, smoothness, and related properties of real-valued functions.

It also includes measure theory, integration theory, Lebesgue measures and integration, differentiation of measures, limits, sequences and series, continuity, and derivatives. Questions regarding these topics should also use the more specific tags, e.g. .

145439 questions
26
votes
3 answers

Does strict convexity imply differentiability?

I know that convexity does not imply differentiability, for example f(x)=|x| is convex but not differentiable. However, |x| is not strictly convex. So I wonder whether strict convexity imply differentiability. I did some search and found out the…
rudi
  • 261
26
votes
1 answer

Function $f:[0,1] \to [0,1]$ taking on each value in $[0,1]$ exactly twice

I want to find a function $f:[0,1] \to [0,1]$ such that $f$ takes on each value in $[0,1]$ exactly twice. I think this means there are an infinite number of discontinuities. Can anyone help me figure this one out? Anyone have any pointers?
Jackson Hart
  • 1,600
25
votes
5 answers

Proving that there exists an irrational number in between any given real numbers

Possible Duplicate: Density of irrationals I am trying to prove that there exists an irrational number between any two real numbers a and b. I already know that a rational number between the two of them exists. My idea was to say represent a and…
Hyperwhatt
  • 251
25
votes
5 answers

Approximation to $ \sqrt{2}$

I'm a first year Undergraduate student from India. Our professor is going to start a Real Analysis course in September and I was preparing for the initials. I tried and solved many problems, but this one has me confused. Probably the main reason for…
Gaurav Tiwari
  • 1,541
25
votes
5 answers

How to prove that derivatives have the Intermediate Value Property

I'm reading a book which gives this theorem without proof: If a and b are any two points in an interval on which ƒ is differentiable, then ƒ' takes on every value between ƒ'(a) and ƒ'(b). As far as I can say, the theorem means that the fact ƒ'…
NonalcoholicBeer
  • 7,211
25
votes
0 answers

No maximum(minimum) of rationals whose square is lesser(greater) than $2$.

Suppose $A$ is the set of all rational numbers $p$ such that $p^2 <2$ and $B$ is the set of all rational numbers $p$ such that $p^2 > 2$. We want to show that $A$ contains no largest element and $B$ contains no smallest element. In Rudin's…
NebulousReveal
  • 13,935
24
votes
1 answer

Creating a question that use the $\epsilon$-$\delta$ definition to prove that $f$ is a continuous function

Let $f:\Bbb R\backslash \{1 \} \to \Bbb R$ be defined by $f(x)= \frac{1}{(1-x)}$. Use the $\epsilon$-$\delta$ definition to prove that $f$ is a continuous function. I do not need answers for it. I want your help to twist the questions a little bit…
user114351
  • 241
24
votes
3 answers

If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$

If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$ Since $(a_n)$ is decreasing and bounded below, by Monotonic Convergence Theorem, $(a_n)$ converges. So, there exists $N$ such…
Idonknow
  • 15,643
24
votes
3 answers

Why isn't Dominated Convergence Theorem taught in intro analysis

In a course based off a book like Rudin's Principles of Mathematical Analysis that does non-measure theoretic analysis why isn't dominated convergence taught? It would be useful since continuous functions are Lebesgue measurable. Is there not a way…
Chris Z
  • 860
24
votes
4 answers

Why does changing variables work?

I am slightly ashamed to be asking this, but I have been recently reflecting on changing variables in very simple problems. If I missed a question that already discusses this please point it out to me and I will delete this one. Anyhow writing this…
Monolite
  • 4,433
  • 5
  • 24
  • 52
24
votes
3 answers

In $\mathbb{R}^n$, locally lipschitz on compact set implies lipschitz

I need to prove: Let $A$ be open in $\mathbb{R}^m$, $g:A \longrightarrow \mathbb{R}^n$ a locally lipschitz function and $C$ a compact subset of $A$. Show that $g$ is lipschitz on $C$. Can anyone help me?
Kenchin Haos
  • 271
23
votes
1 answer

Bibliography for Singular Functions

I wound up assembling a rather lengthy and partially annotated bibliography for my answer to the math StackExchange question Singular continuous functions, but it seems I got a little too carried away and found myself with a post well over the…
Dave L. Renfro
  • 36,843
23
votes
7 answers

Show that the sequence $\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},...$ converges and find its limit.

Show that the sequence $$\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},...$$ converges and find its limit. I put the sequence in this form , $(x_n)$ where $$\large x_n=2^{\Large\sum_{k=1}^{n}\left(\frac{1}{2^k}\right)}$$ I want to use Monotonic…
Idonknow
  • 15,643
23
votes
4 answers

Definition of $L^0$ space

From Wikipedia: The vector space of (equivalence classes of) measurable functions on $(S, Σ, μ)$ is denoted $L^0(S, Σ, μ)$. This doesn't seem connected to the definition of $L^p(S, Σ, μ), \forall p \in (0, \infty)$ as being the set of measurable…
Tim
  • 47,382
23
votes
11 answers

What is the length of a point on the real number line?

Since an interval is made up of an infinite number of points, I am considering the relation of the length of an interval and the length of a point, this lead me to ask what is the length of a point on the real number line ? The nested interval…
iMath
  • 2,237
1 2
3
…
99 100