## Tagged Questions

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### Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$. In the book "Understanding Analysis" by Abbott is …
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### Principal value of integral

What is the principal value of the integral $$\int \limits _0^\infty \left( \frac {1}{x^2}-\frac{\cot(x)}{x} \right) dx ?$$ Maple finds $PV \int_0^\infty \tan(x)/x dx = \pi/2.$ Su …
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### Cauchy’s left endpoint integral (1823)

An old problem: to show that every bounded left integrable function is also Riemann integrable. I know some (for me) not elementary proofs: Gillespie and Kristensen et al (theorem …
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### Irrationality of the integral

Does there exist any rational $t$, such that $|t|<1/10$ and the integral $$\frac{1}{\pi^2}\int\limits_0^{\pi t}\arccos{\left(\frac{\sin{s}}{1+2\sin{s}}\right)}ds$$ is irrational …
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### Newton integration without integration

Consider a function f continuous on a compact interval. Approximate it by a sequence of polygonal functions (you can). Then consider a sequence of primitives of the polygonal fun …
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### Fourier series expansion of Tanh(cos(x))

I need the coefficients of the Fourier series expansion of $\tanh(\alpha cos(x))$, for every real positive number $\alpha >0$ namely, since it is an even function, I need the comp …