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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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If the product of two continuous functions is zero, must one of the functions be zero?

Suppose that I have two continuous functions $$f : \left[ a, b \right] \rightarrow \mathbb{R} \quad \text{and} \quad g : \left[ a, b \right] \rightarrow \mathbb{R}$$ and they have the following property $$f(x) \times g(x) = 0 \space , \forall x…
Mike
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Domains of continuity

I was playing around with the definition of uniform continuity, and realized that a nice application of it is the possibility to extend functions. For example, suppose we are given a uniformly continuous function $f:\mathbb{Q}\to\mathbb{R}$. By…
the L
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Supremum of a Continuous Function is Continuous

I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out. Let $f$ be a continuous function on $[a,b] \subset \mathbb{R}$. Define $f^\star (x)$ as: $$ f^\star(x) =…
Roger Burt
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Any function $f:\mathbb{R} \to \mathbb{R}$ is sum of two Darboux functions

From Wikipedia: Darboux functions are a quite general class of functions. It turns out that any real-valued function f on the real line can be written as the sum of two Darboux functions. This implies in particular that the class of Darboux…
ShakesBeer
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Does absolute convergence of a sequence imply convergence?

In my real analysis notes I've got that absolute convergence of a real SERIES implies convergence of the series. However what about absolute convergence of a sequence? Does this imply convergence of the sequence?
Spongebob Squarepants
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Why, intuitively, does $log(x)$ come in as the integral of $1/x$, wheras the integral of other powers of $x$ are powers of $x$?

Question in title really, something I always found strange when I was learning calculus. I can see that $\int \frac{1}{x} dx$ can't be $\frac{x^0}{0}$ since this is not defined, and then the definite integral $\int_1^t \frac{1}{x} dx$ comes down to…
noname
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Functions of Bounded and Unbounded Variations

I have two questions, I would like to be helped in. Here they are: Show that $$f(x) = \begin{cases} x^2\sin\left(\frac{1}{x^2}\right) &\mbox{if } x \neq 0\\ 0 & \mbox{if } x = 0. \end{cases} $$ is not of bounded variation on $[-1,1]$. Show that…
Nana
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Math Analysis Courses online

Can somebody recommend me respectable Math Analysis courses online? I am a student and I took real analysis course in my university, but I am unsatisfied with the quality of that course. I am even considering switching university because I cannot…
Marina
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Limits of series, proof of the convergence of two sequences

I have two sequences $x_i$ and $y_i$ defined by their expressions : $$x_i-x_{i+1}=y_i-y_{i+1}=\sqrt{x_{i+1}y_{i+1}}$$ I have to prove that $xy(x-y)=0$. I tried this : I have $$x_i=x^{2^{i-1}}\prod_{j=0}^{j=i-2}…
JamelG
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If $f,g$ are simple measurable functions, show that $f+g$ and $f\,g$ are too

This homework problem involves showing that if $f,g$ are measurable simple functions, then so is $f+g$ and $f\,g$ - without using: 1) $\{x \in A: (f+g)(x) < t\} = \bigcup_{r\in\mathbb{Q}} \left[ \{x\in A: f(x) < r\} \cap \{x \in A: g(x) < t-r\}…
nate
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The total variation and the integral of the derivative

I need a hint (not a complete solution) of the following problem: EDIT: when I was posting the question, I found I suddenly got it, so I need some verification. Suppose $F$ is a complex-valued function of bounded variation on $[a,b]$. Then the…
Yai0Phah
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Must an uncountable subset of R have uncountably many accumulation points?

This question is taken from problem 4.1.8 of "Real Analysis and Foundations" by Krantz The question reads: "Let S be an uncountable subset of $\mathbb{R}$. Prove that S must have infinitely many accumulation points. Must it have uncountably…
Upside
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Curve Selection Conjecture

I have the following conjecture which I cannot seem to settle either way: Let $f:[0,1]\to\mathbb R^2$ be a differentiable function such that $f(0)=(0,0)$. Then there exists a continuous function $g:[0,1]\to\mathbb R^2$ such that: 1) $g(0)=(0,0)$ 2)…
caffeinemachine
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Integral with respect to greatest integer function

Assume $f$ is continuous on $[1,n]$. How would you go about taking the integral $$\int_1^n f(x)\,d\lfloor x\rfloor$$ where $\lfloor x\rfloor$ represents the greatest integer function?
user10311389
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Open map as a corollary of the inverse function theorem

Let $U \in \mathbb{R}^n$ be a open set and $f:U \rightarrow \mathbb{R}^n$ be a continuously differentiable function such that $Df(x_0)$ is an isomorphism for every $x_0 \in U$. I'm trying to use the inverse function theorem to show that $f(U)$ is a…
u1571372
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