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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. .

145439 questions
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Differentiable functions such that the derivative is nowhere continuous.

Is there a function $f:\mathbb R\to\mathbb R$ which is differentiable but such that the derivative is nowhere continuous?
idm
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weak convergence implies boundedness.

I have these in books without proof, mostly as a corollary. I was wondering if I could get a proof. Suppose $$\lim_{n\to \infty} \int_0^1 f_ng dx = \int_0^1 fg dx$$ for all $g\in L^2(0,1)$, where $f_n, f \in L^2(0,1)$. Then there exists a…
Josh
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If $\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$, prove $f$ is identically $0$

$f:[0,1] \to \mathbf R$ is continuous. If $$\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$$ for all $x$ in $[0,1]$, prove that $f$ is identically $0$. My thought is to prove that the maximum and minimum of $f$ are equal then $f$ is constant and this constant…
user236626
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Construct a function that takes any value even number of times.

I'm looking for a continous function $f: [0,1] \to \mathbb{R}$ such that it takes any value even (thus finite) number of times. I suppose that it exists in the class of Lipschitz functions. All my approaches have led to some value which is taken…
Pasha Mishchenko
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Condition to guarantee $f=0$ on $[a,b]$

I have been stuck for several days on this old Analysis problem (I am doing some study on my own). I have tried several things (which I'll indicate below), but I cannot seem to figure it out. Here is how the problem is presented: Problem: "Let $f$…
user252043
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does there exist a discrete set whose image is dense

I want to know whether my proof is correct or not : Does there exist a descrete set whose image is dense in $S^1$ under the map $e^{2\pi ix}$ from $\mathbb{R}\rightarrow S^1$? my attempt is : We know that there is a 1-1 correspondence between $S^1$…
Myshkin
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$f$ is positive continuous function on $[0,1]$.

$f$ is positive continuous function on $[0,1]$. Define $$\int_{0}^{a_n} f(x) dx = \frac{1}{n} \int_{0}^1 f(x) dx$$ where $a_n>0$. Find $ \lim_{n\to \infty} n a_n$. It is clear that $lim_{n\to \infty} a_n =0$ because $f(x)$ is positive.I tried to…
mp100
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Well defined measure

Let $X$ be a nonempty set and let $\mathcal{M}$ be the $\sigma$-algebra of countable subsets and cocountable subsets of $X$. Let $\mu(A) = 0$ if $A$ is countable and 1 if $A$ is cocountable. I want to prove that $\mu$ is a measure, but I'm worried…
change_picture
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Show that if $f$ has compact support, then its Fourier transform cannot have compact support unless $f=0$.

Let $f\in$$L^1(\mathbb{R})$ and $g(y)=\int_{\mathbb{R}}f(x)e^{-iyx}dx$. Show that if $f$ has compact support, then $g$ can not have compact support unless $f=0$. First, I assume that $g$ has compact support. Since $f$ has compact support, there is…
Zank
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convergence with respect to integral norm but not pointwise

I want to give an example of a sequence of functions $f_1 \dots f_n$ that converges with respect to the metric $d(f,g) = \int_a^b |f(x) - g(x)| dx$ but does not converge pointwise. I'm thinking of a function $f_n$ that is piecewise triangle, whose…
ben_lee
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Roots of a power series in an interval

Let $ a_0 + \frac{a_1}{2} + \frac{a_2}{3} + \cdots + \frac{a_n}{n+1} = 0 $ Prove that $ a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n = 0 $ has real roots into the interval $ (0,1) $ I found this problem in a real analysis course notes, but I don't even…
Lin
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Show constructively that the sequence definition of continuity implies the epsilon-delta definition

As I understand it, continuity of a real valued function f at a point x can equivalently be defined in terms of sequences or in the epsilon-delta way. Sequence Definition: For every Cauchy sequence $\{x_i\}$ converging to x, $\{f(x_i)\}$ is a…
Smithey
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uniform limit of step function

Define a step function to be a function that is piecewise constant, $$ f(x)=\sum_{i=1}^{n}c_i\chi_{[a_i,b_i)},$$ where $[a_i,b_i)$ are disjoint intervals. Prove that every continuous function on a compact interval is a uniform limit of step…
arthur
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$ |f''(x)+2xf'(x)+(x^2+1)f(x)|\leq1 $ for all $x$. Prove $ \lim_{x\rightarrow\infty }f(x)=0$

Let $f(x):(0, \infty)\rightarrow \mathbb{R} $ be a twice continuously differentiable function such that | $ f''(x)+2xf'(x)+(x^2+1)f(x) |\leq1 $ for all $x$. Prove $ \lim_{x\rightarrow\infty }f(x)=0$ I think this can be solved by applying …
user229886
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Proving function is $C^k$

This question is from an exercise in Way of Analysis (section 10.2.4 problem 20). If $f: \mathbb{R} \rightarrow \mathbb{R}$ is $C^k$ and $f$ is even, then show $F: \mathbb{R}^n \rightarrow \mathbb{R}$ given by $F(x) = f(|x|)$ is also $C^k$. $C^k$…
John C
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