Extrapolate data to a straight line

I have data

with ListPlot I reach the plot:



but the blue one is the exctraced from data. The desired curve is red one in the plot. How I can extrapolate to reach the red one? However, plot before 0.5 is OK albeit if be smoother is better.

You can make a fit. What part of the data you use, depends on the application and the quality of the data. Without further information, I would take data between 0.5


Of course, you may have other preferences to choose the data.





EDIT: If you add a requirement that the Piecewise function be continuous