Whether we talk about EEG, ECG or EKG aren't all these signals discrete as we are reading the voltages from the electrodes that are due to certain bio-chemical or bio-electrical activities. I do not get why people say that EEG is continuous and hence we need to convert it to digital when it is already digital.
Biological signals are analogues and hence continuous. Early EEG systems simply recorded the analogue signals and displayed it as a continuous signal graphically as wiggly lines written by little pens on a roll of paper (Fig. 1). Only after analogue-to-digital converters became available, could EEG signals and the likes be digitally sampled.
Fig. 1. Analogue EEG signal. source: Britton et al., 2016)
To add to the answer provided by @AliceD, pure digital waveforms are square waveforms as they represent steps between 1s and 0s and are therefore not continuous.
Analogue waveforms are not. They are smooth continuous waves and can represent many voltage points at each millisecond, microsecond or nanosecond between the peaks and troughs.
Outputs from ADCs (Analogue to Digital Converters) will have steps in them representing the various voltage ranges at their relative timings. In a sense they are digital outputs but are really a digitized version (a representation) of an analogue waveform through digital sampling of the wave voltage at the relevant millisecond, microsecond or nanosecond depending on the ADC timing (see the image below from https://www.allaboutcircuits.com/textbook/digital/chpt-13/practical-considerations-adc-circuits/).
I disagree with the other answers, to an extent. Biological signals are generally created by ion passing through a cell membrane. Each of these ions cause a tiny current/magnetic field which is too small to measure in most cases. Each cell action potential is made up of hundreds (???) of ions passing through the membrane at a variety of positions. Most measures of biological signals measure potentials from many cells. These cells are distributed in a non homogeneous extracellular matrix. At the limit the biological signals are discrete and made up of individual ions, but by the time we measure them, they are so mixed up that they are continuous.