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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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Integral of $\sin(e^t)dt$

Let $f(x) = \int_x^{x+1} \sin(e^t)dt$. Show that $$ e^x|f(x)| < 2 $$ and that $$ e^xf(x) = \cos(e^x) - e^{-1}\cos(e^{x+1}) + r(x) $$ where $|r(x)| < Ce^{-x}$ for some constant $C$. I can do the first part easily. Integration by parts gives…
Noah Halford
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Completion of the space of piecewise-constant functions on $[0,1]$

Let $L$ be the space of piecewise-constant functions on $[0,1]\subset \mathbb{R}$ equipped with the supremum norm (i.e. step functions). What is the completion of this space? We discussed in my class that all metric spaces have (unique) completion,…
user381606
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Taylor series of $\tan x - \tan (\sin x)$ has all coefficients positive. Why?

It's well known that $x > \sin x$ for $x> 0$. The Taylor series of $ x - \sin x$ is also well known, and the coefficients are alternating. However, it appears that the Taylor coefficients of the function $\tan x - \tan (\sin x)$ are all positive (…
orangeskid
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Do we really need axioms to define order in $\mathbb R?$

I apologize if this question appears to be a dumb one. However given my preliminary knowledge in real analysis, I am unable to resolve the issue; it’s about the order axiom (of reals). It can be shown that algebraic properties of reals follows from…
s a
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Rolle's Theorem

Let $f$ be a continuous function on $[a,b]$ and differentiable on $(a,b)$, where $a
drawar
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question about derivative of exponential function

When I proved derivation the exponential function expose with problem that have to use derivative of $e^x$ $$\frac{de^x}{dx} = \lim_{h\to 0}\frac{e^{x+h} -e^x}h=\lim_{h\to 0} e^x \frac{e^h-1}h =e^x \cdot \lim_{h\to 0} \frac{e^h-1}h$$ Calculate …
maisam hedyelloo
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$f=\infty$ on a set of measure 0, then $\int_E f = 0$

Let $E$ be a set of measure zero and define $f = \infty$ on $E$. Show that $\int_E f = 0$. This is out of Royden 4E, p 84. I know how to prove this if $f=0$ on $E$. But I'm curious, as stated, won't this result in a situation in which $\infty \cdot…
emka
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Can we create a dense set in the interval by this steps?

I have a question that is something I am wondering for some time now and I couldn't even begin answering it. I guess it could be called a riddle. So, let $x\in[0,1]$ and $k\in(0,1)$. We begin at $x$ and we say that we make a "step up" by going to…
tst
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Let $f:[0,\infty)\to \mathbb R$ be Continuous and strictly decreasing function such that : $\lim_{x\to \infty}f(x)=0$

Let $f:[0,\infty)\to \mathbb R$ be Continuous and strictly decreasing function such that : $\lim_{x\to \infty}f(x)=0$ prove $$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}$$ Diverges. I can not prove. please help .
Almot1960
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graph is dense in $\mathbb{R}^2$

I was asked in a exam: does there exist a function(need not be continous) $f:\mathbb{R}\rightarrow \mathbb{R}$ whose graph is dense in $\mathbb{R}^2$? I proved that graph of a discontinuous linear map is dense but did not provide explicit example,…
Myshkin
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Union of fat Cantor sets?

A question came up asking me to find two disjoint sets $A, B$ such that $[0, 1] = A \cup B$, $A$ is meager and $m(B) = 0$. My thought was the following: Let $\mathcal{C}_k$ denote the fat Cantor set obtained, starting with $[0, 1]$, by removing the…
dasaphro
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Prove $e^n$ and $\ln(n)$, mod 1, for $n=2,3,4...$ is dense in $[0,1]$

How can one prove $e^n$ and $\ln(n)$, modulo 1, are dense in $[0,1]$, for $n=2,3,4...$? By dense is meant, for any $0
fff
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Number of real solutions.

Question : Let $\{a_i\}$ be a sequence of real numbers such that $0
Random-generator
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If $x_n \geq 0$ for all n $\in N$ and $\lim((-1)^nx_n)$ exists. Show that $x_n$ converges.

If $x_n \geq 0$ for all n $\in N$ and $lim((-1)^nx_n)$ exists. Show that $x_n$ converges. Let $\lim((-1)^nx_n)=l$ therefore, for $\epsilon>0$ $\exists k\in N $ such that $| (-1)^nx_n - l|<\epsilon/2$ $\forall n\geq k$ $\implies |x_n + l| <…
vishu
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How to construct a bijection from $(0, 1)$ to $[0, 1]$?

Possible Duplicate: Bijection between an open and a closed interval How do I define a bijection between $(0,1)$ and $(0,1]$? I wonder if I can cut the interval $(0,1)$ into three pieces: $(0,…
ymfoi
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