- #1
coverband
- 171
- 1
How come sin(x^-1) is not continuous and xsin(x^-1) is?
coverband said:In the sequential definition of a limit, n approaches infinity and we show there exists a natural number, say p, such that n>=p implies |f(n)-L| < epsilo
Do you mean to prove sin(x^-1) is not continuous we show we cannot find a real number r, such that x<=r implies
|sin(x^-1) -0|< epsilon ?
HallsofIvy said:At some points larger than 1/delta, you will have values of y such that sin(y)= 1 and some at which sin(y)= -1. That means that for 0< x< delta, you will have some values of x so that sin(1/x)= 1 and some for which sin(1/x)= -1. As soon as you take epsilon< 1/2, you can't have |sin(1/x)-L|< epsilon for any delta. If |1- L|< 1/2, |-1- L|= |1+ L| can't be less than 1/2.
In case the point hasn't been made clearly, NEITHER of these functions is continuous at x= 0. x sin(x^-1) has a removable discontinuity at x= 0. The function defined by "f(x)= x sin(x^-1) if x is not 0: and "f(0)= 0" is continuous for all x but that is no longer the same function.coverband said:How come sin(x^-1) is not continuous and xsin(x^-1) is?
Continuity analysis is a method used in research and data analysis to examine the changes and patterns in a particular phenomenon or event over time. It involves studying the consistency and flow of data points or observations to identify trends, gaps, and potential disruptions.
Continuity analysis can be applied to a wide range of data, including numerical, categorical, and qualitative data. It can also be used to analyze both quantitative and qualitative variables, such as time series data, survey responses, and textual data.
The main steps in conducting a continuity analysis include identifying the research question, selecting appropriate data and variables, cleaning and organizing the data, visualizing the data using graphs or charts, and interpreting the results to draw conclusions.
Continuity analysis differs from other data analysis methods, such as regression analysis or trend analysis, in that it focuses on the overall flow and consistency of data over time, rather than specific relationships or patterns. It also allows for the detection of potential disruptions or changes in trends, rather than just analyzing existing trends.
Continuity analysis has various applications in different fields, such as economics, social sciences, and environmental studies. It can be used to analyze market trends, population changes, policy impacts, and natural disasters, among others. It can also be used in forecasting and decision-making processes to identify potential risks and opportunities.