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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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Prove that meas$(A)\leq 1$

Let $f:\mathbb R\to [0,1]$ be a nondecreasing continuous function and let $$A:=\{x\in\mathbb R : \exists\quad y>x\:\text{ such that }\:f(y)-f(x)>y-x\}.$$ I've already proved that: a) if $(a,b)$ is a bounded open interval contained in $A$, and…
uforoboa
  • 7,074
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Perfect set of irrationals

I solved the following exercise: Let $\{r_1,r_2,r_3,\dots \}$ be an enumeration of the rational numbers and for each $n \in \mathbb N$ let $\varepsilon_n = 1/2^n$. Define $O = \bigcup_{n=1}^\infty V_{\varepsilon_n}(r_n)$ and let $F=O^c$. (a) Argue…
newb
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Uniform convergence of piecewise linear interpolations

Let $$X^k (t) := X^0 (t+t_k) - X^0 (t_k)$$ where $X^0(t)$ is the piecewise linear interpolation of $X^0(t_k) \equiv X_k=\sum_{i=0}^{k-1} a_i b_i$ with interpolation intervals $a_k$. $t_k=\sum_{i=0}^{k-1} a_i$, $t_0=0$. So we can think of supporting…
Johannes L
  • 699
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Does "continuous + only at countable many points not differentiable (with bounded derivative)" imply Lipschitz-continuity?

Let $f$ be continuous on $\mathbb R$ and differentiable with derivative $f'$ on $\mathbb R \setminus \{t_0, t_1, \dots \}$. Let $\sup | f'(t) | < \infty$, then $f$ is Lipschitz continuous with $L=\sup |f'(t)|$. Does this hold? How could one prove…
Johannes L
  • 699
4
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2 answers

$L^1([0,1])$ is not a Hilbert space.

Show that $L^1([0,1])$ is not a Hilbert space. The problem, as it is originally stated (here at http://orion.math.iastate.edu/dept/grad/analysis_f06.pdf), implies that we should show the parallelogram law fails for some potential inner product…
Darrin
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Finite Union of Compact Sets Clarification

While studying some basic analysis/topology I have come across the proof regarding that the finite union of compact sets is compact using the definition of compactness. The proofs I have read all basically follow this: For each compact set choose a…
user7090
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Limits of a function and its derivative

Let $u:[0,\infty) \to \mathbb{R}$ be a continuously differentiable function in $t$, and let $$t^{n-1} u'(t) + \frac{1}{2} t^n u(t) = C$$ for some constant $C$ and positive integer $n$. Suppose that $\displaystyle\lim_{t\to +\infty} u(t) = 0$ and…
user1736
  • 8,573
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A question about functions in $L^p\big([0,1]\big)$

Let $f \in L^1 ([0,1]).$ Prove that for each $0<\alpha<\frac{1}{2}$, $$\int_0^1 \left( \frac{\left|f(x)\right|}{x} \right)^\alpha dx$$ is finite. I attempted to use Jensen's Inequality, but there is, of course, no guarantee that…
Darrin
  • 3,155
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product of two continuous proper maps

Let $f,g: \mathbb{R} \to \mathbb{R}$ be continuous proper maps (i.e., $f^{-1}(K)$ is compact for all $K \subset \mathbb{R}$ compact). Is $fg$ proper?
user117315
  • 53
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prove that if $f(x)+xf'(x)>0$ then $f(x)>0$

I would appreciate if somebody could help me with the following problem Q: prove that ($f:\mathbb{R}\to \mathbb{R}$) if $f(x)+xf'(x)>0$ then $f(x)>0$
Young
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Proving the map is jointly continuous

Let $f: \mathbb R \times \mathbb R \to \mathbb R$ be the map $(x,y) \mapsto xy$. For my own exercise I wanted to prove it is continuous. It seemed easy to assume that the topology on $ \mathbb R \times \mathbb R$ is generated by the $\max$-norm.…
blue
  • 2,884
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Cauchy-Schwarz application

Let $f:(0,\infty) \rightarrow \mathbb{R}$ be continuous. I need to show that $$\left(\int_1^ef(x)dx \right)^2 \leq \int_1^e xf(x)^2dx$$ I have been trying to use C-S to prove this but with no luck.
user9352
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Absolutely continuous functions. Prove a limit.

I need help to prove this. Let $f:\mathbb{R}\longrightarrow{\mathbb{R}}$ be an absolutely continuous function in any interval of the form $\left. [ -k,k \right ]$. If $f^{\prime}$ is in $L(\mathbb{R})$ and $\left\{{x_n}\right\}$ is a sequence that…
user95747
  • 261
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4 answers

prove inequality through mean value theorem

I want to prove that if $x$ is a real number and $x>-1$, $$\sqrt{1+x} \leq 1+\dfrac{x}{2}$$ I'm not sure which one should I choose for $f(x)$. I also don't understand why we need the mean value theorem to prove this.
Kingkong
  • 163
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$f$ satisfies $|f(x)-f(y)|\ge \frac{1}{2}\cdot|x-y|$ , is $f$ onto?

Suppose $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $|f(x)-f(y)|\ge \frac{1}{2}\cdot|x-y|$ for all $x,y\in \mathbb{R}$, is $f$ onto? I know this is true if $|f(x)-f(y)|\ge |x-y|$
Mathronaut
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