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Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...

2 votes

Properties of slowly-varying functions at two large points

(I will use a definition of a slowly varying function from Wikipedia) No, of course not. Here is an example: Let us first of all for convenience switch to the additive formulation by considering $g(t) …
Aleksei Kulikov's user avatar
5 votes
Accepted

A generalized logarithmic function

Let us calculate $f_{\epsilon}(x) - \frac{1}{\epsilon}\log(1 + x)$ using your formula for $\log(1 + x)$ as $f_1(x)$(I didn't checked it but believe that it is correct): $f_\epsilon(x) - \frac{1}{\eps …
Aleksei Kulikov's user avatar
4 votes
Accepted

If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $...

Sure you can find such a $g$ and without much difficulty. Without loss of generality we assume that $f(0) = 0$ (otherwise just subtract it). Construct the sequence $\delta_n>0$ inductively in such a w …
Aleksei Kulikov's user avatar