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Here are the finishing times taken for a race in seconds.

$$10.1,10.6,11.2,12.1,10.9,11.3$$

Is this data discrete or continuous?

My understanding is that time is continuous data but these data are presented in a to-one-decimal-place form.

Are they now discrete?

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    The set of possible values is continous, hence the data is continous as well. The values occuring in practice are discrete because we cannot determine them exactly, so we have to round at some point. – 2017-01-27
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    The measurements indeed define a discrete variable, as the stopwatch provides a single decimal (or a single decimal is noted), and you have no knowledge of the continuous time. – 2017-01-27

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The concept of 'continuous data' is a theoretical convenience. In practice, all data must be rounded to a certain number of decimal places.

Theoretically, running times are continuous. One can imagine infinitely many values between 10.1 and 10.2, even if the measurement device is not up to measuring them. Maybe the winner between 10.1 and 10.2 would be determined by a photograph. Whether you model these six observations as continuous or discrete may be a matter of tradition or convenience.

How will you plot the data? Perhaps a dotplot or stripchart, focusing attention on the six individual discrete values observed.

enter image description here

Or perhaps you might use a histogram, speculating on the theoretical continuous distribution that may have produced these data. Six observations are not nearly enough for this to be successful, but below we show the "best-fitting" normal density (blue curve) and a kernel density estimator (green), just to give an idea of what might be done with more observations. The tick marks beneath the histogram show the six observations.

enter image description here