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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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There exist $x_1, x_2, x_3$ such that $\frac{1}{f'(x_1)} + \frac{1}{f'(x_2)} + \frac{1}{f'(x_3)} = 3$

Let $f$ be a real-valued function defined in $[a, b] \subset \mathbb{R}$, with $f(a) = a, f(b) = b$. Suppose that $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Show that there exist three distinct points $x_1, x_2, x_3$ such…
user182973
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Does there exist a function$f: [0,1]\rightarrow \mathbb{R}$ almost everywhere $0$ but whose range is equal to $\mathbb{R}$

Does there exist a function$f: [0,1]\rightarrow \mathbb{R}, f=0,a.e$ but whose range is equal to $\mathbb{R}$? I can't image what this kind function looks like.
Shine
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$f:[0,1] \rightarrow \mathbb{R}$ is absolutely continuous and $f' \in \mathcal{L}_{2}$

I am studying for an exam and am stuck on this practice problem. Suppose $f:[0,1] \rightarrow \mathbb{R}$ is absolutely continuous and $f' \in \mathcal{L}_{2}$. If $f(0)=0$ does it follow that $\lim_{x\rightarrow 0} f(x)x^{-1/2}=0$?
Mykie
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Is the distance function differentiable?

Let $I=[0,1]\subset \mathbb{R}$ and for $x\in \mathbb{R}$ define $f(x)= \operatorname{dist}(x,I)$ then I need to find out its points of differentiability. I could see that $f(x)$ is continuous everywhere, but for differentiability at some point…
Mathronaut
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How can I visualize the discrete metric?

I saw that in a real analysis proof, they used a proof by contradiction where the metric was a discrete metric. That is, distance is defined to be 1 if the points ARENT the same and 0 if the points are the same. I was trying to visualize how this…
user123276
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$f(x,y)$ is such that partial derivative w.r.t $x$ is zero, but$ f$ still depends on $x$?

I have a problem where it seems like I should be able to visualize an answer, but I can't. Perhaps I need to take a more formal approach. "Let $A$ be a non-empty open convex subset of $\mathbb{R}^2$, and suppose that $f: A \rightarrow \mathbb{R}$…
user153568
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$L^1$ function unbounded on every interval

[My question concerns part of Exercise 2.25 in Folland's Real Analysis text.] I'm looking at the function $g(x)=\sum_{n=1}^\infty 2^{-n}f(x-r_n)$, where $f(x)=x^{-1/2}$ for $x\in (0,1)$ and $f(x)=0$ elsewhere, and $\{r_n\}$ is some enumeration of…
1234
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Analyzing Integral of a product using Cauchy Mean Value Theorem?

Prove that if the functions $g,h:[a,b] \rightarrow \mathbb{R}$ are continuous, with $h(x) \geq 0$ for all $x\in[a,b]$, then there is a point $c$ in $(a,b)$ such that $\int_a^bh(x)g(x)dx=g(c)\int_a^bh(x)dx$. At first I tried to use…
pmal
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Does $f$ exist such that $f(1)<0 , f(5)>3 $ and $ f'(x)\le e^{-f(x)}$

Does there exist a continuously differentiable function $f:[1,5]\rightarrow\mathbb{R}$ such that $f(1)<0 , f(5)>3$ and $f'(x)\le e^{-f(x)}$ ? My Attempt : If such $f$ exists the mean value theorem states: $\exists c \in(1,5)…
uqtredd1
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Limits of Monotone Functions

I've been studying about limits of functions using Introduction to Analysis by Gaughan. A few days ago I asked this question Limits of Functions about the limits of functions. The motivation was curiosity about whether the idea of proving that a…
CritChamp
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Prove that there can be at most countably many disjoint letter T's in the plane

A letter T in the plane is defined as a non-zero length segment with an orthogonal non-zero length segment that has an end-point in the strict interior of the first segment. Prove that there can be at most countably many disjoint letter T's in the…
user2566092
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Fejér's Theorem (Problem in Rudin)

Can you solve Problem 19 from Chapter 8 of Rudin's Principles of Mathematical Analysis, I'm having a lot of difficulty with it I've proven the first part, namely $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N…
Stephen
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Prove functions defined by sup and inf are continuous

Suppose $f$ is continuous on $[a,b]$. Show that the functions defined by $m(x)=\inf\{f(y):y\in[a,x]\}$ and $M(x)=\sup\{f(y):y\in[a,x]\}$ are well defined and are also continuous on $[a,b]$ I have already managed to prove that they are well defined,…
mjh
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About derivatives of real function

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function, $A \subset \mathbf{R}$ be a closed set and $x_0, y \in \mathbf{R}$. Assume that for each $\varepsilon >0$ there exists $\delta >0$ such that if $x \in A$ , $|x-x_0|< \delta$…
Richard
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Theorem 6.11 of Rudin's Principles of Mathematical Analysis

In the proof of Theorem 6.11, $\varphi$ is uniformly continuous and hence for arbitrary $\epsilon > 0$ we can pick $\delta > 0$ s.t. $\left|s-t\right| \leq \delta$ implies $\left|\varphi\left(s\right)-\varphi\left(t\right)\right|<\epsilon$. However,…
tediso
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