# Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with limits, convergence of sequences, construction of the real numbers, least upper bound property and related analysis topics, such as continuity, differentiation, and integration through the Fundamental Theorem of Calculus.

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### Is $f(y)=\chi\!\left(\bigcup\limits_{k=1}^{\infty}\left[q_{k},q_{k}+\frac{1}{4^{k}}\right]\right)(y)$ Riemann-Integrable on $\mathbb{R}$?

Is $$f(y)=\chi\!\left(\bigcup\limits_{k=1}^{\infty}\left[q_{k},q_{k}+\frac{1}{4^{k}}\right]\right)(y), \hspace{0.2cm} \{q_{k}\}_{k \in \mathbb{N}} \subseteq \mathbb{Q}$$ Riemann-Integrable ? ...
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### Why $f_n=\frac{1}{n}1_{[0,n]}$ shows that $L^1(\mathbb R)$ is not reflexive?

In my course it's written that $f_n=\frac{1}{n}1_{[0,n]}$ shows that $L^1(\mathbb R)$ is not reflexive. Could someone explain why ? I know that reflexive mean that bounded sequence has subsequence ...
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### $f'(x_0)\geq \frac{f(x_0)-f(x)}{x_0-x}$?

Assuming that $f\in\mathcal{C}^2(\mathbb{R})$ and $\max f'(x)=f'(x_0)$, then why $$f'(x_0)\geq \frac{f(x_0)-f(x)}{x_0-x} ?$$ It might be a silly question, but I'm stuck. Thank you in advance.
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### Conditions for Differentiable Limit Theorem

I know that if $(f_n)$ is a sequence of differentiable functions on $(a,b)$ with pointwise limit $f$ and if $f'_n \rightarrow g$ uniformly the $f$ is differentiable on $(a,b)$ and $f' = g$. I want to ...
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### A sequence $x_n$ has a cauchy subsequence ii it has a subsequence satisfying the following property [closed]

I am trying to solve this problem but could not make an idea. Please give some hint for the problem.
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### Proving limits using $\varepsilon$-$\delta$ definition. How to verify that computed $\varepsilon$ definition is correct?

So from our textbook exercises I was practicing on, it says that we need to prove this limit is true using epsilon delta definition. $\lim_{(x,y)\to(0,2)}\ -3x +4y = 8$ I've came up with the result ...
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### Limit question multivariable analysis

Let f be a map from $\mathbb{R}^2$ to $\mathbb{R}^2$ be defined as $f(x,y)=(x^2,y^2)$ let T from $\mathbb{R}^2$ to $\mathbb{R}^2$ be the map $T(x,y)=(2x,4y)$ then to show that f is differentiable at ...
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### The integral $\int_{0}^{\infty} e^{-x^5}dx$ is convergent or not?

So, my question is whether the integral $\int_{0}^{\infty} e^{-x^5}dx$ is convergent or not? My work- So, I tried to use the comparison test. I see that $\frac{1}{e^{x^5}} \leq \frac{1}{x^5}$ but ...
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### Solution of a differential equation hitting a constant - possible?

In my ODE lecture, we came across the DE $$\frac{dy}{dx} = y(1-y)$$ I know that the solution to this is simple to derive but there are also 2 constant solutions $y \equiv 0$ and $y \equiv 1$. I'm ...
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### If $f'(c)>f'(x)$ for all $x\in(c-\epsilon,c)\cup(c,c+\epsilon)$, then $f'(c)>\frac{f(x)-f(c)}{x-c}$ for all $x\in(c-\epsilon,c)\cup(c,c+\epsilon)$? [closed]

Is the above claim true? If yes, how might we prove it? If no, what is a counterexample?
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### Why do they use $\frac{1}{n}$ balls in this proof for closed sets?

I can follow the proof all the way upto the fact that we need to show that no $B_\epsilon(x)$ lies completely in $T$ (thus the contradiction later). If no ball lies completely in $T$, then it must ...
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### Use the deﬁnition of the limit of a sequence to establish the following limits [closed]

1) $\lim_{n\to\infty}\frac{\sqrt n}{n+1} = 0$ 2) $\lim_{n\to\infty}\frac{(-1)^nn}{n^2 + 1} = 0$
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### Prove or disprove that there does not exist a monotone function $f:\mathbb{R}\rightarrow\mathbb{Q}$ which is onto.

Prove or disprove that there does not exist a monotone function $f:\mathbb{R}\rightarrow\mathbb{Q}$ which is onto. Clearly $f$ can not be continuous. Suppose $f$ is discontinuous. Then it can have ...
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### Let f, g be defined on R and let c ∈ R, suppose that lim x->c f(x)=b, and that g is continuous at b, show that lim x-> infinity (fg) = g(b) [duplicate]

have no idea, hope for any help Let f, g be defined on R and let c ∈ R, suppose that lim x->c f(x)=b, and that g is continuous at b, show that lim x->infinity (fg) = g(b)