can pure differentiators be implemented in digital control?

I think not. Digital implementations are always based on differences. Differences are taken between samples taken at different times. Instantaneous differentiation is not possible, only an approximation.

Approximate differentiators are. But the differentiation always has an upper frequency roll off. Otherwise, the gain would have to be infinite at infinite frequency. Opamps have a gain that rolls off at higher frequencies.

I reject the premise. It is not difficult to differentiate with an op-amp, provided one has reasonable expectations. Obviously, for any given scaling, there is a slope whose representation will saturate the circuit, but analogous limitations also occur with integrators and linear amplifiers.

Differentiation is not an operation that can be performed with discrete-time calculation. We approximate it with differences, which is not quite the same thing.

Jerry

No, a pure differentiator cannot be implemented in any sampled-time scheme. Moreover, you don't want a pure differentiator in any real control loop -- the unbounded gain at high frequencies is very undesirable, because the phase shift of the plant gets very dicey up there. Wise implementers of differentiators always make them lead-lag filters, in analog or digital realms.