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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. .

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Continuous function maps $F_{\sigma}$ sets to $F_{\sigma}$ sets

Prove if $X\subset \Bbb{R}$ is $F_{\sigma}$ (can be written as a countable union of closed sets) and $f$ is continuous then $f(X)$ is $F_{\sigma}$. Proof: Let $X=\cup C_i$ where $C_i$ is closed. Then define $D_{i,n}=C_i \cap [-n,n]$. Then…
user223391
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Lebesgue's Differentiation Theorem for Continuous Functions

If $f:\mathbb{R}^n\to\mathbb{R}$ is continuous, does Lebesgue's differentiation theorem hold at all points? That is, does $$\lim_{r\to0}\frac{1}{|B(x,r)|}\int_{B(x,r)}f(y) \, dy=f(x)$$ $\textit{everywhere}$?
Dan Burrows
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Is it possible to interchange order of supremum and supremum?

Prove that if A and B are arbitrary sets and f is a bounded real-valued function on $A\times B$, then $$ \sup_{a \in A} \sup_{b \in B} f(a,b) = \sup_{b \in B} \sup_{a \in A} f(a,b) . $$ If it is possible, then does it change into one supremum like…
Dkdg
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Differentiable+Not monotone

Is there a real function that is differentiable at any point but nowhere monotone?
t.k
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Solve the equation: $e^x=mx^2$

I need to find out the maximum possible number of real roots of the equation: $$e^x=mx^2$$ where m is a real parameter. I'm interested in some easy approaches. Moreover, is it possible to solve it without using derivatives at all? Thanks.
user 1591719
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Prove that exist a function $f:\mathbb{R}\rightarrow \mathbb{R}$ infinitely differentiable.

Prove that exist a function $f:\mathbb{R}\rightarrow \mathbb{R}$ infinitely differentiable, such that $$\int_{\mathbb{R}}f(t)\,dt=1\;\mbox{ and }\; \int_{\mathbb{R}}t^nf(t)dt=0,\, \forall\, n\geq 1\; \mbox{integer}.$$ Some ideas? Thank you.
ElliptCg
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Locally lipschitz on $[a,b]$ implies lipschitz

Suppose a function $f:\mathbb{R}\rightarrow\mathbb{R}$ is locally Lipschitz. Prove that $f$ is Lipschitz on $[a,b]$. Here is what I have so far: Let $[a, b]$ be some closed, bounded interval. Since f is locally Lipschitz, for each $x\in[a; b]$, we…
john
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Does there exist a continuous onto function from $\mathbb{R}-\mathbb{Q}$ to $\mathbb{Q}$?

Does there exist a continuous onto function from $\mathbb{R}-\mathbb{Q}$ to $\mathbb{Q}$? (where domain is all irrational numbers) I found many answers for contradicting the fact that there doesnt exist a continuous function which maps rationals to…
Shubham Ugare
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Difficult Intermediate Value Theorem Problem- two roots

I have been stuck on this Real Analysis problem for hours and am just totally clueless- I am sure it is some application of the Intermediate Value Theorem- suppose $\ f: \mathbb{R}\rightarrow\mathbb{R} $ is continuous at every point. Prove that the…
MathNYYB
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Counterexample for Interchange of Limits in integration

If $f_n$ converges to $f$ uniformly in $\mathbb{R}$, then \begin{equation*} \lim_{n\to\infty}\int_a^b f_n(x)\,dx =\int_a^b f(x)\,dx \end{equation*} but it's not true in general that \begin{equation*} \lim_{n\to\infty }\int_{-\infty}^\infty…
Prayagdeep Parija
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Is the series $\sum \sin^n(n)$ divergent?

I'm almost sure that the series $\sum \sin^n(n)$ is not convergent, but lack proof. Thank for any help.
Luciano Battaia
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If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$

Let $f:\mathbb{R}_+ \to \mathbb{R}_+$ be a monotone decreasing function defined on the positive real numbers with $$\int_0^\infty f(x)dx <\infty.$$ Show that $$\lim_{x\to\infty} xf(x)=0.$$ This is my proof: Suppose not. Then there is $\varepsilon$…
Galois
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Finding an integer $n$ such that $\sin(n)$ is close to 1

Given some $\epsilon>0$, is there an efficient way to find an integer $n$ such that $$1-\sin(n)<\epsilon$$ We all know there is always one (and many), and so I can test all $n$ from $0$ until I find a good candidate, but I ask for some efficient…
Xoff
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If $f(x) = h(x)g(x)$, is $h$ differentiable if $f$ and $g$ are?

I know that if I have two differentiable functions $f, g$ then the functions $(f + g)$ and $fg$ are also differentiable. I would like to find a way how to argue about the function $h$ where \begin{equation} f(x) = (hg)(x) := h(x)g(x) \quad…
harlekin
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Given $f(0)>0, f(1)<0$ then prove $\exists x_0$ st $f(x_0)=0$

Let $f$ be a function satisfying $f(0)>0, f(1)<0$ , prove that $\exists x_0$ st $f(x_0)=0$ under the assumption that there exists continuous function $g(x)$ such that $f+g$ is non decreasing. I noticed that if $0\le x\le1$ then $$g(0)
Mathronaut
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