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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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How do I show this $\cos^2(\arctan(x))=\frac{1}{1+x^2}$

How do I show this? $$\cos^2(\arctan(x))=\frac{1}{1+x^2}$$ I have absolutely no idea. Thank you.
user185346
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Is a function with non-negative first partial derivatives increasing?

Suppose $f(x_1,x_2,\dots,x_n)$ is a multivariable function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that for all partial derivatives, $1\le i \le n$, $$\frac{\partial f}{\partial x_i}(q_1,q_2,\ldots,q_n) \ge 0$$ for all $q_i \ge r_i$. Is f a…
Henry B.
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Is it possible to make R^n a field for n>2

I know that real line is a field. And we can make $\mathbb R^2$ a field by defining complex operations. But can we extend this process? Is there any way to make $\mathbb R^n$ field, at-least for some positive integer '$n$'? In particular, using…
mathman
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Is $f(x,y) = (x^3 + xy^3) / (x^2 + y^6)$ differentiable at 0?

Is $f(x,y) = (x^3 + xy^3) / (x^2 + y^6)$ (for $x \neq 0$ or $y \neq 0$), $f(0,0) = 0$ differentiable at 0? Answer: No. Proof: f is not continuous at 0: If we approach 0 via $x = y^3$, the limit of $f$ equals $1/2$. Thus $f$ is not…
CHwC
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4 answers

Defining dense subsets of $\mathbb{R}$

Here is a quote from Derbyshire's "Unknown Quantity" The rational numbers are "dense." This means that between any two of them you can always find another one. This being a pop. math book, this is not meant as a formal definition of dense sets but…
nolion
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$\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$

1.Let f be a real-valued differentiable function defined on [0, 1]. If $f(0)=0$ and $f(1)=1$, prove that there exists two numbers $x_1,x_2 \in [0, 1]$ such that $\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$. 2.Let f be a real-valued differentiable…
Benji
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Deriving 2-variable Taylor expression from 1-variable

Suppose we know Taylor's theorem for $f: \mathbb{R} \rightarrow \mathbb{R}$ with remainder. How would we use this to derive Taylor's theorem for $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ with remainder? To start, I'm trying to figure out which…
countunique
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Does the proof that the Nested Set Theorem implies the Axiom of Completeness require the Archemedean property?

Here is the beginning of a standard proof that the Nested Set Theorem implies the Axiom of Completeness (e.g. see here and here): Suppose $E$ is a non-empty set bounded above by $b$ and let $a$ be an element of $E$. Define the following sequence…
David
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Trying a Calculus question

One of my pals asked me to look at this question Let $f: [0,1] \to \mathbb{R}$ be differentiable. Suppose that $f(0) = 0$ and $0 < f'(x) < 1$ for all $x \in (0, 1)$, where $f'(x)$ is the derivative of $f$. Prove that $$ \left(\int_{0}^{1}f(x) \ dx…
picakhu
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Is intermediate value property equivalent to Darboux property?

I always thought that a function $f:\mathbb{R} \to \mathbb{R}$ has the intermediate value property (IVP) iff it maps every interval to an interval (Darboux property): Proof: Let $f$ have the Darboux property and let $a
Andrei Kh
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Arzela-Ascoli Theorem: Is only pointwise boundedness required?

In Royden's text the Arzela-Ascoli Theorem states: Let X be a compact metric space and $f_n$ a uniformly bounded, equicontinuous sequence of real valued functions on X. Then $f_n$ has a subsequence that converges uniformly on X, to a continuous…
qwert4321
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Proving a bounded monotone function has finite one-sided limits

all! I got stuck on this question today, although it seemed straight forward when I started. Here is the proposed problem: Let a and b be extended real numbers with a $\lt$ b. Prove that if f is a bounded, monotone function on the interval (a, b),…
rlh282
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Lipschitz Implies $F(x)=F(0)+\int_0^x F'(t) dt$.

I've been working on the following question: If $F : \mathbb{R} \rightarrow \mathbb{R}$ is a Lipschitz function, then $F(x)=F(0)+\int_0^x F'(t) dt$. I've already proved that Lipschitz implies $F'$ is exists a.e., and $F'$ is essentially bounded, but…
Frank White
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My proof uses derivatives. Can I extend it to non-differentiable functions?

For every function $f:\mathbb{R}^+\to\mathbb{R}$ and positive constants $a,b$, define a function $f_{a,b}:(0,1)\to\mathbb{R}$: $$ f_{a,b}(x) := f(2a\cdot x) + f(2b\cdot(1-x))$$ A function $f$ is good if the function $f_{a,b}$ has a unique maximum…
Erel Segal-Halevi
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Dense subset of $C[0,1]$

In $C[0,1]$ the set $\{f(x): f(0)\neq 0\}$ is dense? I know only that polynomials are dense in $C[0,1]$, could any one give me hint how to show this set is dense?thank you.
Myshkin
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