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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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I want to make sure my proof is correct using $\epsilon, \delta$ definition of continuity for $\sin\left(\frac{1}{x}\right)$

So, let's consider the function $$ f(x)=\begin{cases} \sin\left(\frac{1}{x}\right) & x\neq 0 \\ 0 & x=0 \end{cases} $$ I want to prove the above function is discontinuous at $x=0$. Let $\epsilon=\frac{1}{2}$. And let's consider…
Miguel Duran Diaz
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Proving that $X$ is a metric space

If $X$ is the set of all sequences $(x_k)_k$ such that $sup_{k\in N}|x_k|$ exists, then with $$d(x,y)=sup_{k\in N} |x_k-y_k|$$ is $X$ a metric space? So, for this I have to prove the three properties of $d$ hold? I'm having trouble proving these,…
user23899
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Injective function $f : \mathbb R \to \mathbb R$ that is dense in $\mathbb R^2$

Problem for fun, not homework. Is there an injective/bijective function $f : \mathbb R \to \mathbb R$ that is dense in $\mathbb R^2$? I am completely lost as to how to begin on this sort of problem. I was able to prove the existence of a surjective…
A.S
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Use mean value theorem to find $f(\xi)+f''(\xi)=0$

Question: $f\in C^2(-\infty,+\infty)$, $|f(x)|\leq 1$, $[f(0)]^2+[f'(0)]^2=4$. Prove that there exists $\xi$ s.t. $$ f(\xi)+f''(\xi)=0. $$ I find this question in the chapter about mean value theorem, and I set up function $G=f^2+f'^2$ as usual.…
yahoo
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What is your definition of Riemann-Stieltjes Integral?

I'm studying Rudin-PMA and the definition here is different from that of Wikipedia. Rudin PMA page 122 Wikipedia Rudin says the condition that "integrator should be monotonically increasing and integrand should be bounded on $[a,b]$" is necessary to…
Katlus
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$f'$ is bounded $\Rightarrow$ $f$ is bounded?

Let $I$ be a bounded connected subset in $\mathbb{R}$ and $f: I\rightarrow \mathbb{R}^k$ be a differentiable function. Does boundedness of $f'$ imply boundedness of $f$? (I edited this post after I realized that I didn't actually write what i…
Katlus
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Checking if a set is closed / open

I have the set $A = \{\frac{1}{n} + \frac{1}{k} \mid n,k \in \mathbb{N}\} \subseteq \mathbb{R}$. My exercise asks me to find the closure of this set, but my question in this post is simply asking for clarification on how open sets work. A set is…
user463756
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Example of continuous function on closed and unbounded set in $R$ with no maximum

What is an example of a continuous function on a closed and unbounded set with no maximum? Is $f(x)=x^3$ a correct example?
Maximiliano
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Proof involving limits of functions and their derivatives

I'm working on another analysis problem: Let $f$ be a differentiable function on an interval of the form $(a,+\infty)$. Prove that if there is a number $r > 0$ such that $\lim_{x\to\infty}(rf′(x)+f(x))=L$ is finite, then $\lim_{x\to\infty}f′(x)=0$…
themathandlanguagetutor
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Cauchy Schwarz for integrals (as Spivak would have it done)

One of the first problems in Spivak's Calculus on Manifolds asks you to prove the Cauchy-Schwarz inequality for real integrable functions, namely, that $|\int_{a}^{b}fg|^2 \leq |\int_a^bf^2||\int_a^bg^2|$. Now, the easiest way I see of doing this is…
Duncan Ramage
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If $h(r, \theta) = f(r \cos \theta, r \sin \theta)$, show that $ f_{xx}+f_{yy} = h_{rr} + \frac{1}{r} h_r + \frac{1}{r^2} h_{00}$

If $h(r, \theta) = f(r \cos \theta, r \sin \theta)$, show that $$ f_{xx}+f_{yy} = h_{rr} + \frac{1}{r} h_r + \frac{1}{r^2} h_{00}$$ Hint: Rewrite the defining equation as $f(x,y) = h(r(x,y), \theta(x,y))$, with $r(x,y) = \sqrt{x^2+y^2}$ and…
gegu
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$\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists?

The question is the following: Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, and for all $x \neq 0$, $f'(x)$ exists. If $\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists? My intuition is that this does not…
Kalypso
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Image of countable and uncountable sets

There are two questions: Image of a countable set of real numbers under any continuous function is countable? My claim is yes. Let $X$ is countable $\implies X=\{x_1,x_2,\ldots,\}$. Now $f(X)=\{f(x_1),f(x_2),\ldots,\}$ which can be atmost…
Sachchidanand Prasad
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Spivak's Calculus 8-3(a)

This question is from Spivak's Calculus (3rd ed) Chapter 8 on Least upper bounds. Let $f$ be a continuous function on $[a,b]$ with $f(a)<0
helios321
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Existence of integral implies the existence of a limit

Suppose that $f$ is a decreasing continuous function on $[0, \infty)$. And the integral of $f(x)/\sqrt{x}$ on $[0, \infty)$ exists. Prove that $\lim_{x\rightarrow \infty}\sqrt{x}f(x)=0$. My work. I think we should prove the limit exists firstly.…
Intoks Liobein
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