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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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In mean value theorem, does the mean value vary continuously?

Let $f\colon\mathbb R\to\mathbb R$ be continuously differentiable and let's say, for simplicity, that $f(0)=0$. Then by mean value theorem it's $$f(x)=f'(\xi)\cdot x \,\text{ for some } \xi \in (0, x)$$ What I wondered is: What can we tell about the…
Dario
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Sequence of bounded functions or pointwise bounded

We have sequence of function $\{f_n(x)\}$ defined on set $E$ and $n\in \mathbb{N}$. What does mean sequence of bounded functions? Is it pointwise bounded? Definition: We say that $\{f_n\}$ is pointwise bounded on $E$ if the sequence $\{f_n(x)\}$ is…
RFZ
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Question about the proof of Rudin's Theorem 2.30

The theorem states: Suppose $Y \subset X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E = Y \cap G$ for some subset $G$ of $X$. I think the proof in the forward direction is relatively clear, however I have some problems relating…
user118837
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why the equation $f^{(n)}(x)=0$ has at least $n-1$ distinct roots in $(-1,1)$

Let $f \in C^{(n)} ( (-1,1) )$ and $\sup_{-1
Destiny freedom
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Application of Stone-Weierstrass Theorem

Suppose that $f \colon [0,1] \rightarrow \mathbb{R}$ is a continuous function on $[0,1]$ with $$\int_0^1 f(x)\ dx = \int_0^1 f(x)(x^n+x^{n+2})\ dx$$ for all $n=0,1,2, \dots$. Show that $f\equiv 0$. Can someone help me with this question? Is this…
KWO
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Let $f : \mathbb{R} → \mathbb{R}$ be continuous, with $f(x)f(f(x)) = 1$ for all $x ∈ \mathbb{R}$. If $f(1000) = 999$, find $f(500)$.

Let $f : \mathbb{R} → \mathbb{R}$ be continuous, with $f(x)f(f(x)) = 1$ for all $x ∈ \mathbb{R}$. If $f(1000) = 999$, find $f(500)$. I try to solve this problem, but I don't know how to use de continuity on $f$. Is anyone could give me a little…
user230283
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Is every function $f$ on $ \mathbb R^2$ such that $f(x,y) \le g(x) + g(y)$ for every $(x,y)$, for some function $g$ on $\mathbb R$?

Is the following statement true? For every $f: \mathbb R^2 \to \mathbb R$, there exists $g:\mathbb R \to \mathbb R$, such that $f(x,y) \le g(x) + g(y)$ for all $x,y \in \mathbb R$. I do not think so. However, I couldn't find a counterexample. In…
user258700
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$f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.

I am not sure how to do this. I can prove it if I know $f$ is bounded, but otherwise I am stuck. $f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.
user24900
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Lebesgue Integral Questions

I'd like to show some properties of the Lebesgue integral. I'd like to show that if $f$ is a simple function which is zero almost everywhere, then the Lebesgue integral $\int f(x) dx = 0$. Similarly, I'd like to show this is also true for a…
J. Chong
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Possible mistake in Royden and Fitzpatrick's book Real Analysis

Proposition 19(v) of Section 1.5 page 23 in the book Real Analysis by Royden and Fitzpatrick, 4th edition (see link), says: If $a_n \le b_n$ for all $n$, then $\lim\sup a_n \le \lim \inf b_n$. However one can find a counter-example for a sequence…
user79963
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Theorem 4.22 from baby Rudin

Can anyone explain me what would be if one of them is empty?
RFZ
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Rudin's PMA: Exercise 2.24

I have some difficulties solving the following exercise (Rudin's Principles of Mathematical Analysis (PMA), 2.24) Let $ X $ be a metric space in which every infinite subset has a limit point. Prove that $ X $ is separable. In order to solve this I…
user251106
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Prove that $f$ has a fixed point.

Let $f:[0,\infty [\to[0,\infty [$ continuous such that $$\lim_{t\to\infty }\frac{f(t)}{t}=\ell\in[0,1).$$ Prove that $f$ has a fixed point, i.e. there is an $x\geq 0$ such that $f(x)=x$. I don't really know how to solve this problem. My first…
idm
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Monotonic function only has jump discontinuities

I'm trying to show that a monotone function on a closed interval can only contain jump discontinuities. Could someone give me a hint as to how I should begin? I am not sure how to start this problem. Edit: Let $f$ be a increasing function. Then if…
Student
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Theorem 12.13 of Apostol's Mathematical Analysis (2nd Ed)

I had a little difficulty understanding Theorem 12.13 of Apostol's Mathematical Analysis, and was wondering if someone who had read this book could give me some pointers. Specifically, on page 360, the author writes "As a consequence of Theorem…
syeh_106
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