Data like this is often called quantized, particularly when the numbers' precision is limited by the measurement device. For example, a scale might only display integer numbers of grams or pounds. This is particularly common when an analog signal (from a microphone, strain gauge, etc) is digitized. The resulting error (e.g., the difference between 0.012 and 0 for your first data point) is called quantization error. You could also call it rounding or discretization, though this faintly implies that it was done during post-processing.
Truncation also works here, but one needs to distinguish between truncating the range of the observations (e.g., converting anything above 10 into 10, or below 0 to 0) and truncating the values of individual observations.
I'm not aware of a way to robustly detect quantization in any situation. In fact, pretty much all data is quantized to some extent and the amount of quanitization is often known ahead of time from the measuring device's specifications. However, there are some easy heuristics you could try:
How many unique values do you have? Digital-to-analog converters use a fixed number of bits (typically 8, 12, 16, or 24), which gives you $2^8, 2^{12}, 2^{16}$ or $2^{24}$ unique values, and these values are often equally spaced between the maximum and minimum value.
Is there a consistent step-size between the values. In other words, sort them, throw out duplicates, and see if the neighboring values typically increase by the same amount.
Still, I think you'd be better off inquiring about how the data was generated to begin with.
If the data is "mildly" quantized, it's usually not an issue. For example, I wouldn't worry too much if my human subjects' weights were recorded in (integer) pounds or kilograms. If the data is heavily quantized, you could treat it as interval-censored data. This is particularly common in survival analyses, where you might only check to see if someone is alive or something is functioning at some fixed interval (e.g., weekly inspections of a factory). Search for interval regression if this fits your situation.
You should be sure to understand the null hypothesis underlying any tests you run on binned data. For example, data uniformly distributed across 10 bins is quite different from data uniformly distributed across the entire range.