# 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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### Finding example of a function having the required property.

Does there exist any continuous function $f : \Bbb R \longrightarrow \Bbb R$ which is not differentiable only at the integers and is not uniformly continuous everywhere? The only function which I can ...
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### Prove that $a<b$ iff $a^3<b^3$ (don't use difference of two cubes)

I would like to know a different proof for this problem: Let $a,b\in \Bbb R$. Show that $a<b$ iff $a^3 < b^3$. Given that it is not much difficult to prove by using a difference of cubes and ...
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### Solving Basel Problem using euler infinite product and infinite product-sum equality

Trying to prove Basel problem through the equality $sin(x) = \prod\limits_{k=1}^{+\infty}(1-\frac{x^{2}}{\pi^{2}k^{2}})$, I came across the following problem; I was able to prove the following ...
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### Well ordered subsets of $\mathbb{R}$

At the entrance exam of a french school, the following problem was given : Characterize well-ordered subsets of $\mathbb{R}$ The only property I found was that such a subset must be at most ...
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### How to tell whether a function has a root in a specific set?

There is a smooth function $$f:\mathbb{R}\rightarrow\mathbb{R}$$ Suppose that we know everything about behaviour of this function $f(x)$ for $x\ge M$ where $M$ is real number. To know everything i ...
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### Struggling to understand why two different definitions of Baire Category theorem are same

I have two versions of Baire Category theorem and I am struggling to find why they are equivalent: My professor notes says " Any complete metric space is of second category i.e. we cannot write it as ...
Let $0<a<1$ be arbitrary but fixed. The equation $$\frac{x^2 (1+y)^2}{\left(a y + \sqrt{1+(1-a^2)y}\right)^2} = 1$$ in $y$ has according to straight-forward calculus and Mathematica two ...