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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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On the existence of a continuous bijection $f\colon [0,1]\to [0,1]\times [0,1]$

Let $f$ be a continuous function on $[0,1]$ such that $f([0,1])=[0,1]\times[0,1].$ Then show that $f$ is not one-one. Hints will be appreciated.
Mirunalini_UML
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$\int_0^1|f(x)-g(x)|dx=1$ for distinct $f,g\in S$

Does there exists an infinite subset $S$ of $C([0,1],\mathbb{R})$ such that $$\int_0^1|f(x)-g(x)|dx=1$$ for any distinct $f,g\in S$? I was guessing the the answer is yes. I can construct such a set with 3 functions, but can't really be…
naoh
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Prove that $\int_0^1 \frac{dx}{f^2(x)+1} \le \frac{ \pi}{4}$

Let $f:[0,1] \to \mathbb{R}$ be a differentiable function, for which $f'(x) \ge 1 , \forall x\in [0,1]$, and $f(1)=1$. Prove that: $$\int_0^1 \frac{dx}{f^2(x)+1} \le \frac{ \pi}{4}$$ From the hyphotesis, we deduce that $f(0) \le 0$. This doesn't…
npatrat
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Definition of the Derivative Using a Sequence

I believe that I may have once seen a definition of the derivative that went along these lines: $$f'(c)=\lim_{n\to\infty}\frac{f(x_n)-f(c)}{x_n-c}$$ Here, $(x_n)$ is a sequence that converges to $c$. As I do not recall much about it, I may have…
wjmolina
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True or False: If sets $A$ and $B$ have a maxima, and $A \cap B \neq \emptyset$, then $A \cap B$ has a maxima

I am almost certain that the statement in the title is True, but am not 100% sure how to prove it, or if my conclusion is valid. My reasoning is that since $A$ and $B$ both have a maxima, then they have an upperbound which belongs to their…
elbarto
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Fact in proof of Lebesgue's Differentiation Theorem

I'm reading a proof of Lebesgue's Differentiation Theorem, where there is a fact that is not further specified. Let $f \in L^1(\mathbb R^n)$. For $r > 0$ we set $$f_r(x) := \frac{1}{\lambda^n(\mathbb B(x, r))} \int_{\mathbb B(x, r)} f(y) \, \mathrm…
aexl
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Recursive Sequence Limit

I'm trying to show that the limit of the following recursive sequence is 4. $a_{n+1} = \frac{1}{2}a_{n} + 2$ and $a_{1} = \frac{1}{2}$. Could someone give me a hint as to how to start this problem? I've been stuck on this for a while.
user26139
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Axiom of Completeness to prove intermediate value theorem

I am having a little trouble understanding one of the steps in this proof. From Stephen Abbott's Analysis: Using AoC to prove the IVT: TO simplify matters, consider $f$ as a continuous function which satisfies $f(a)<0
elbarto
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If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function, $f(0)=0$ and $f' = f^2$, then $f = 0$.

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function, $f(0)=0$ and $f' = f^2$, then $f = 0$. Any help?
violeta
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Prove that the Interior of the Boundary is Empty

Suppose X is a Metric Space Let S $\subset X$ Prove that if S is Closed then, the Interior of the Boundary of S is Empty Totally stuck on how to solve this.
user145003
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Proving every positive natural number has unique predecessor

I am independently working through Tao's Analysis I, and one of the exercises is to prove that every positive natural number has a unique predecessor. The actual lemma is (where $n++$ denotes the successor of $n$): Let $a$ be a positive [natural]…
russell11
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If $F: \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ is continuous and $\| F(x) - F(y)\| \geq \lambda \| x - y \|$ is $F$ a surjection?

In my real analysis class my professor gave us the problem of proving that if $F: \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ is continuous and satisfies $\| F(x) - F(y)\| \geq \lambda \| x - y \|$ then $F$ is a bijection with continuous inverse.…
user127562
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Limit of the Derivative of an Increasing, Bounded-Above Function

Let $f:(0,\infty) \to \mathbb{R}$ be a differentiable function, which is increasing and bounded above. Then does $\lim_{x \to \infty} f'(x)=0$? If we assume that $\lim_{x \to \infty} f'(x)$ exists, then this is true by an argument using the mean…
user4601931
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Lagrange/Cauchy mean value theorem

Let $\,\,f,g\in C^2([0,1])$ such that $f'(0)g''(0)-f''(0)g'(0)\neq 0$ and $g'(x)\neq 0$ for all $x\in (0,1)$. Let $\theta(x)$ be a real number such that $$\frac{f(x)-f(0)}{g(x)-g(0)}=\frac{f'(\theta(x))}{g'(\theta (x))}.$$ What can I say about…
Felice Iandoli
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Trying to understand the concept of limsup and liminf of sets

Let $(E_n)$ be a sequence of sets. I was giving the following definitions: $$ \limsup_{n \to \infty} E_n = \bigcap_{k=1}^{\infty} \bigcup_{n \geq k} E_n $$ $$ \liminf_{n \to \infty} E_n = \bigcup_{k=1}^{\infty} \bigcap_{n \geq k} E_n $$ I am having…
user203867
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