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Just to add to what @Kostas VI. mentioned: Measuring means always approximation. If the approximation is "good enough", you would consider it "reliable data". What that exactely means has to be decided in each individual context.
As general rule: if the data has higher resolution than the minimal size you need for your task, it's probably "good enough". A resolution of 1 meter in a Digital Elevation Model (DEM) is probably good enough to identify buildings (thus "continuous enough"), but might be not ideal if you want to identify structures on the rooftop (like chimneys, roof structure and so on).
If you have a DEM, taken either from satellite (like SRTM) or via aerial survey (LIDAR data), you get point data (or point clouds) - every sample point with it's own hight. Than, these raw data is processed in such a way as to generate a continuous surface (interpolation) - thus a "model".
These models often come in the form of a (Geo) Tiff, thus a digital raster image. The very concept of raster images implies that they are always discrete: they consist of pixels. But depending on technical resolution and the resolution you need for your task, there are cases where you could consider them as almost continuous, even if in fact they are not.
Also consider the coastline paradox: you can always find a better resolution for the thing you want to measure of capture in an image. The more you go into the details, the more "structure" you find.
But normally, a certain resolution is high enough for what you want to do with the data. In this case, if the technical resolution of the measurement used is higher, than you can consider the data to be "almost continuous" in a broader sense, even if from a technical view, digital instruments always save data in a discrete form (the very concept of digital technologies is based on this).
To make a long story short: there is no clear answer to your question, it depends an what kind of data you have and for what you need them: thus - what in your context the meaning of "continuous" and "discrete" exactely means (in a technical sense, to repeat, digital data is always discrete).
I think it depends on what you want to do with it and what the image depicts. For example, a satellite image depicting Ocean Color (e.g. Sentinel-3 OLCI product) or Sea Surface Temperature (SLSTR product) can be considered continues, since the variables they depict are inherently continuous variables. However, if you wanted to discriminate between some distinctive ocean features (e.g. streams, eddies) you could classify them based on some rule, thus consider them as discrete.
If you're talking about higher spatial resolution land surface data (e.g. Landsat data) and you want to make a land cover map, then you need to consider them as discrete, i.e. different class corresponds to a different surface. If you want to distinguish between surface objects it doesn't make any sense to consider them as continuous. For example, if there is a highway road next to a crop, you'll take them as discrete.
But, if for some reason you want to take the spectral values of each pixel and do something more generic with it (e.g. a heatmap of spectral values), you could e.g. smooth the image bands.
Another example: Imagine you have Landsat data. Imagine you make a random cross-section of a spectral band which passes over a crop, building. What would you expect to see in various spatial resolutions? If the spatial resolution was higher then the distinction between the objects would be more clear and the cross-section would have a sudden jump, since building respond differently to radiation than crops. So if you wanted to distinguish between the two, it would make sense to consider them as discrete.