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 Dec17 comment Question involving entire functions Dear pankaj, you have now asked three questions, all of which show no own work. Try investing a little time in the material before you come to us, and if you had any thoughts, please tell us what you've tried. Dec17 comment boundedness of an operator $\frac{f(x)}{1+|x|+|y|}\leq f(x)$ Dec17 comment Show the function $x_1\sin(1/x_2)+x_2\sin(1/x_1)$ is continuous everywhere Okay, so $\lim_{x\to1}F(x,x-1)-(x-1)\sin(1/x)$ should be defined by continuity of F and convergence of the term $(x-1)\sin(1/x)$ (convergence because $x-1\to0$ and $\sin$ is bounded). But this limit is your $x\sin(\frac{1}{x-1})$, which indeed does not converge since $\frac{1}{x-1}$ does not (add a few sentences here), ergo $F$ is not continuous. Dec17 comment Show the function $x_1\sin(1/x_2)+x_2\sin(1/x_1)$ is continuous everywhere I don't know, your reasoning sounds alright but you'll need to add a sentence somewhere - I'm not quite convinced. Dec17 comment Show the function $x_1\sin(1/x_2)+x_2\sin(1/x_1)$ is continuous everywhere The fact that your calculations do not work does not prove $F$ cannot be continuously defined. In particular, the fact that the individual limits do not exist does not prove that the sum does not (for example, consider $0=\lim 1/x - 1/x=\lim1/x-\lim1/x$). Dec17 comment Displaying $2 \cdot 10^5$ as $200000$ Right click on the result, Numeric formatting. Dec17 comment Prove that $\int_0^{+\infty} \frac{\ln x}{a^2+x^2} dx = \frac{\pi\ln a}{2a}$ Perhaps this is not an answer for Leitingok, but this is exactly how they would do it in the book, and shows the power of complex analysis. +1. Dec17 comment Given joint pdf of $X$ & $Y$, find pdf $Z=XY$ Yes. $X\leq1$ and $Y\leq1$ so $Z=XY\leq1$. Dec16 comment $f^{-1}$ and continuity This is actually one of the definitions for continuity, so you really need to explain what definition you are using. Dec16 comment linear hyperbolic PDE with some BCs at infinity Hello there, try putting your equations in latex, as explained here: meta.math.stackexchange.com/questions/107/… Dec16 comment Prove that $f'$ exists for all $x$ in $R$ You make quite a stronger statement than what was asked for, but it's technically correct. Dec16 comment Complex Functions Concept Questions I think the fact that these answers are longer than the normal, mathematical, equation-rich answers, says enough about this method of teaching. Dec16 comment Prove that $f'$ exists for all $x$ in $R$ Dec16 comment Prove that $f'$ exists for all $x$ in $R$ Dec16 comment Prove that $f'$ exists for all $x$ in $R$ Possible duplicate: math.stackexchange.com/questions/64766/… Dec16 comment How to find $f'$ from the definition of derivative? ... so what is $f'(0)$? Dec15 comment How random is the digits of $\pi$? If any of you guys have more requests, I'll be happy to compute/plot more data. Dec15 comment How random is the digits of $\pi$? @PeterSheldrick: mathematica: Histogram[RealDigits[Pi, 10, 10000][[1]], 10] Dec15 comment Find the coordinate matrix of the function $\cos^2x$ relative to the ordered basis $\{1,\cos x,\sin x,\sin 2x\}$. Since when are sets ordered? Dec12 comment In a fraction between integers, what denominators produce a periodic result? Every ratio has periodic decimals, if that's what you're asking.