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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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Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients such that $P(x, \cos x) = 0$ holds for all real $x$

I came across the following real analysis problem while reviewing, and I am genuinely stuck on this one: Show that there is no non-zero polynomial $P(u,v)$ in two variables with real coefficients such that $P(x, \cos x) = 0 $ holds for all real…
Lost in a Maze
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Why it is not a function

This question is very basic one but at this moment I could not answer it. We know $y=\pm\sqrt{x}$ is not a function because one value of $x$ corresponds to more than one value of $y$. Again $y=x^2$ is a function. But we can get the first equation…
Janak
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Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise

Question: Find a sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly to a differentiable function $f$, such that $f'_n$ converges pointwise but not uniformly to $f'$. Attempt: I have tried a number of possibilities,…
mathjacks
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If $\sum{a_k}$ converges, then $\lim ka_k=0$.

I want to prove the following statement: Suppose that $\displaystyle\sum_{k=1}^{\infty}a_k$ converges, where $(a_k)_{k\in\mathbb{N}}\subseteq\mathbb{R}$ is monotone. Then $\displaystyle\lim_{k\to\infty}ka_k=0$. I believe we have several cases. For…
Surtan
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Rudin Principles Theorem 2.40: Every k-cell is compact.

In the proof $I$ is a $k$-cell whose coordinates are bounded by $a_{j}\le x_{j}\le b_{j}$ where $1\le j\le k$. From the proof: Put $c_{j}=(a_{j}+b_{j})/2$. The intervals $[a_{j},c_{j}]$ and $[c_{j},b_{j}]$ then determine $2^{k}$ $k$-cells $Q_{i}$…
MathMajor
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$\int_{0}^{\infty} x \cdot \cos(x^3) dx$ convergence

$$\int_{0}^{\infty} x \cdot \cos(x^3) dx$$ I only want to prove, that this integral converges, I don't need to calculate the exact value. I don't know what to do with the cosinus, I can't get rid of it. I know that the integral is equal to…
Rummelluff
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Show that $f$ is a polynomial if it's the uniform limit of polynomais

Let $f:\Bbb R\to \Bbb R$ be a function which is the uniform limit of polynomials. I want to show that $f$ is a polynomial. I mean this seems a bit trivial... If it's the uniform limit of the set of polynomials doesn't that guarantee it's a…
user146296
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Proof of exponent rule $a^{n/m}$ when $n$ and $m$ belongs in the reals

Now I have not studied math for very long. I have just completed Calculus1, although my knowledge extends a bit outside of this. My question is, how can the transforms below be justified $$ a^{n/m} \, = \, \left( a^{1/m} \right)^{n} \, = \, \left(…
N3buchadnezzar
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If $F$ is strictly increasing with closed image, then $F$ is continuous

Let $F$ be a strictly increasing function on $S$, a subset of the real line. If you know that $F(S)$ is closed, prove that $F$ is continuous.
JimJones
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$f(x)$ and $h(x)$ are absolutely continuous functions. Is $e^{f(x)} |h(x)|$ as well?

Given that functions $f(x)$ and $h(x)$ are absolutely continuous on $[0,1]$, I want to show that $e^{f(x)} |h(x)|$ is absolutely continuous as well. I know that (1) the product of two absolutely continuous function on $[0,1]$ is absolutely…
Alex J.
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Does every inner product space has an orthogonal basis?

It is proved that every inner product space has a basis $W$, but I am not sure if every inner product space has an orthogonal basis? It is known that every inner space has a maximal orthogonal set $S$, so if I want to prove the conclusion, we need…
89085731
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$\int_0^\infty f(x) \; dx < \infty$ implies $\lim_{x \rightarrow \infty} x f(x) = 0$.

Let $f$ be non-negative, monotone decreasing such that $$\int_0^\infty f(x) \; dx < \infty$$ Show that $$\lim_{x \rightarrow \infty} x f(x) = 0.$$ I have the following solution, but wonder if there is a better way. Let $\epsilon > 0$ be small.…
user09812093
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Subset of $l^\infty$ compact or not

I would be grateful if you can give me some hints for the following homework problem. Let $C$ be a subset of $l^\infty$ (with uniform norm) such that $C = \left\{(x_n) \mid |x_n|\leq \frac1n \,\forall n\geq 1\right\}$ Is $C$ a compact set or…
marvinthemartian
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3 answers

Is there a function almost everywhere $0$ on $\mathbb{R}$ whose graph is dense in $\mathbb{R^2}$?

Is there a function almost everywhere $0$ on $\mathbb{R}$ whose graph is dense in $\mathbb{R^2}$? How to establish such strange funciton?
Shine
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If $f_n$ is a sequence of differentiable functions converging to $f$ uniformly on a compact set

Suppose $f_n\rightarrow f$ on a compact set in $\mathbb{R}^n$, with $f_n\in C^1$. $f$ is not necessary differentiable. We can easily find a sequence of functions converging to $|f|$, for example. My question is: does there exist any results which…
Lost1
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