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 Oct 8 awarded Yearling Jan 18 comment Is $\textrm{d}x \in \mathbb{R}$? @user89: I think in part because too many books have been written using standard analysis and too many people have been trained in standard analysis. It would be too hard to throw all those books aside and teach all those people the non-standard approach. Same reason helps explain why Windows is the dominating O.S: you cannot just get rid of all the old programs written for windows and it is not practical to retrain everyone to learn a new O.S. Jan 18 answered Is $\textrm{d}x \in \mathbb{R}$? Jan 18 comment Is $\textrm{d}x \in \mathbb{R}$? For dx as an infinitesimal, it depends on whether the property described is a first-order property, so we can apply the "transfer principle" and elementary equivalence, i.e., the Hyperreals are elementary equivalent to the standard Reals. Jan 18 comment How does $ln(y) = 2x+C$ become $y = e^{2x+C}$? In other words, $e^y$ and $lny$ are inverses of each other with the needed restriction of domains. Oct 8 awarded Yearling Sep 30 awarded Explainer Sep 24 awarded Autobiographer Aug 14 comment Exact interpretration of p-value and significance of test It is always an issue of plausibility; you can never reach absolute certain, but it is more plausible/likely , given the sample data and the distribution, that the mean is actually 100. Aug 14 comment Exact interpretration of p-value and significance of test But the number of observations is factored in the p-value already. Aug 14 comment Exact interpretration of p-value and significance of test Phrasing is confusing. I would say we conclude/assume the mean is 100, but we cannot be 100% sure that this is the case. Aug 14 answered If a property holds for arbitrary compact set in a metric space, does it also holds for the metric space? Aug 14 comment Exact interpretration of p-value and significance of test Right, then you do not reject. But you do not conclude the mean is 100; the sample data may have been either unrepresentative or made up of outliers. You can only conclude that under assumptions (which you cannot guarantee) of data being representative (not outliers ) and randomly-selected, the claim is true. Aug 14 revised Exact interpretration of p-value and significance of test added 381 characters in body Aug 14 comment Exact interpretration of p-value and significance of test Sorry, I misread ;let me add something. Aug 14 comment Exact interpretration of p-value and significance of test Please see the bottom paragraph I just wrote and let me know if it answers your question. Aug 14 revised Exact interpretration of p-value and significance of test added 381 characters in body Aug 14 answered Exact interpretration of p-value and significance of test Aug 14 comment $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping. Are you assuming there is a local homeomorphism in order to use this argument? Aug 14 comment $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping. O.K, so we agree ;).