In which of the following scenarios is it appropriate to use line charts to display the data? I am trying to understand when it is appropriate to use a line chart to display the data in a data set. It is my understanding that the x-variable should either be continuous, or should represent time.
I am comfortable using line charts whenever the time variable is continuous (or near continuous) and the line segments drawn between data points function as some sort of interpolation. However, I am not quite comfortable justifying the use of line charts when the x-variable is time-related, but in a sequential rather than a continuous manner. In those cases it seems like the line segments are there only for increased readability.
I don't know where I can find a definite source that could settle these matters.
In which of the following scenarios would it be inappropriate to use a line chart?
i) The number of pancakes sold in January, February, ..., December.
ii) The temperature at noon on Monday, ...., Sunday
iii) The average temperature on Monday, ...., Sunday
iv) The highest measured temperature on Monday, ..., Sunday
v) The level of rainfall (measured in millimeters) on Monday, ..., Sunday.
vi) The number of hours spent studying on Monday, Tuesday, ..., Sunday.
vii) The number of murders, by year (..., 1989, 1990, ...) .
 A: All your examples are time series, and line charts are appropriate for all of these, as long as all buckets have valid data. (If there is no data for a particular time bucket, then I would not join the two adjacent data points by a line.)

In those cases it seems like the line segments are there only for increased readability.

I would say that increased readability is a perfect reason to use a line chart.
The question always is whether a plot can be misunderstood. If you plot the numbers of murders per year and join the dots by lines, I think nobody would believe that the line indicates something for a hypothetical year "1989.5".
Even if your chart showed "firecracker accidents on New Year's Eve by year", I would still use lines to join dots, because it simply does not make sense to interpret the line joining the 1989 dot to the 1990 as "firecracker accidents on New Year's Eve on June 30, 1990" - because there is no such thing.
In the end, which visualization makes sense always needs to be decided using your context.
