Switching from live plot to fixed plot should set the fixed time range to what was last shown on the live plot. Currently, it shows what happened to be set when the plot was first switched to live mode (could be a range years in the past). This could help when you want to stop the live plot to take a screenshot.
There is a possibility a user would be just poking around and try the live plot, but really does want to return to the fixed plot they had originally. In this case the browser back button should help.
Switching from live plot to fixed plot should set the fixed time range to what was last shown on the live plot. Currently, it shows what happened to be set when the plot was first switched to live mode (could be a range years in the past). This could help when you want to stop the live plot to take a screenshot.
Example